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Page 1: 10(24350(,76&'*#8)&'*29'*):6;'=@?$?BA7C$=@?$?ED · P QSR W D J / W^_QS ... [tFn?h?o R[h ... [hGR[h ¥I`CXZqSR Zn?h?o R[h°o Rk XZnGl

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1

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s\piZtGR[a�\piZQOm�QOt?X.R[h°��q�\piZ� L√

2 = 1.4142135623730950488016887242096980785696718753769480731766 . . .

e = 2.7182818284590452353602874713526624977572470936999595749669 . . .

π = 3.1415926535897932384626433832795028841971693993751058209749 . . .

e

0 1 2 3

2

4

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5�� � 6y8 2{z{|&2p} 8 2Q(√

2) ⊂ R5 �

Ç�a \p�/R[XDa�QOi √2��n�X.R�l]t?hGR[h<¸C�?X]\pnGl.tGR[h���QOX+h?QSl.tFi¦\pqOqSR¤X.R�R[qOqSR[h�r/\pt?qSR[h<Mp® QOXDm�`Ca�abR[h�a�QOi)�FQSR[q_�¤R[h?QOo�R[X

\pnG�@M

6y8�E+7 4 7 0 E 5�� �Q(√

2) := {a+ b ·√

2 | a, b ∈ Q} ⊂ R

��qS�.`1QS�Zi 13 + 12

7 ·√

2R[QOh�i��_Y?QS�.l]tGR��¤g�qSR[abR[hFi \pnG�

Q(√

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g�qSR[abR[hFi.R�\pnG�Q(√

2)��QSR�k?R[XW��n

Q(√

2)o�R[tGVCX.R[h�L

(a+ b√

2) + (c+ d√

2) = (a+ c) + (b+ d)√

2

(a+ b√

2) · (c+ d√

2) = (ac+ 2bd) + (ad+ bc)√

2

© R[abR[XZm�R1�¤R[QOi.R[X@¸kG\C�.�(a+ b

√2)(a− b

√b) = a2 − 2b2.

P1\ √2 /∈ Q

¸?mN\ph?h#� ?Xa, b ∈ Q

k?R[XW��nG�.k�XZnGl]ma2 − 2 · b2 hjn?X 0

�.R[QOh<¸&�¤R[h?ha = b = 0

M&P�R[a���n�� `CqOo�R�/R��ZQOiZ��i �#R�k?R��

a+ b√

2 ∈ Q(√

2)n?h?oNqSR[QSl.t

0R[QOhGR[h�T1R[t?XZ�¤R[XZi�L

(a+ b√

2) ·(

a

a2 − 2b2− b

a2 − 2b2

√2

)

= 1,

\pqS�.`1

a+ b√

2=a− b

√2

a2 − 2b2=

a

a2 − 2b2+

−ba2 − 2b2

√2.

®�R[QOqRR[QOh�T�VCXZY R[X�QS�Zi�n?hGk

Q(√

2) ⊂ R¸?�.R[tGR[h���QOX�\pqS�.` L

= 4 µº5��O³Q(√

2)(H" + &�(*5���� :�� & :G@

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oNqSR[QSl.t −1QS�Zi@M

® QOX�t&\p� R[hb�.l]tG`Chbo�R��.R[tGR[h<¸NkG\C�.�x2 6= 2

� ?X�\pqOqSRx ∈ Q

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2 ∈ Ra�QOi�k?R[X!g�QOo�R[hG�.l.t&\ ��i@¸&kG\C�]�

(√

2)2 = 2.

® QOXb� R��]l.t?X.R[QO�/R[h �ZR[iZ��i�k?R[h¯T�VCXZY R[XC ⊃ R

k?R[X�m�`Ca�Y?qSR�^?R[h r/\pt?qSR[h<M¤gI�boNQO�?i�R[QOhi ∈ C

a�QOi{k?R[Xg�QOo�R[hG�.l]t&\ �9i@¸kG\C�.�

i2 = −1.

© R[abR[XZm�RN¸GkG\C�.�!kG\ph?h¾\pnGl.t(−i)2 = 1

�.R[QOhya�nG�.�@M® QOX�� R[iZX]\Cl]tji.R[h¾hjn?h<¸?oj\ph?� ��X.R�l.t<¸&�WnG�.k�X Gl]m�RUk?R[X!�G`CXZa

z = a+ b · ia�QOi

an?hGk

b\pnG�

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(a+ b · i) + (c+ d · i) := (a+ c) + (c+ d) · i,

(a+ b · i) · (c+ d · i) := (ac− bd) + (ad+ bc) · i,�¤`CX]\pn��Ga�\ph�k�n?X.l.t���nG�Za�n?qOiZQOY?qOQO��QSR[X.R[h�m�`Ca�a�i@¸@n?hji.R[X+© R[X Gl.m��ZQSl.tFiZQOoNn?h?o k?R[XD�WR[o�R[q

i2 = −1M@© R[abR[XZm�RN¸

kG\C�.�(a+ b · i)(a − b · i) = a2 + b2.

¼FQOhGkan?hGk

b\pnG�

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a2 + b2 = 0hFn?X�Y&\C�.�ZQSR[X.R[h<¸ �¤R[h?h

a = 0n?hGk

b = 0M!¼_`Cq�\ph?o�R

z = a+ b · i 6= 0¸?m�VCh?hGR[h���QOXWk�n?X.l]t

zi.R[QOqSR[h�L

1

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a+ bi=

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−1 = i · i =√−1 ·

√−1 =

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1 = 1

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6y8�E+7 4 7 0 E � ��5 ®¯QOX!�.R[iZ�@R[h

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• 0 := (0, 0)k�QSR1m�`Ca�Y?qSR�^?R ��n?qOq M

• 1 := (1, 0)k�QSR1m�`Ca�Y?qSR�^?Rcg�QOhG��M

• i := (0, 1)k�QSRcQOa�\poNQOh��pX.RUg�QOh?tGR[QOi@M

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(a, b) + (c, d) := (a+ c, b+ d)

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(a, b) · (c, d) := (ac− bd, ad + bc)

= 4 µ � � � �E( &�� &�53/N&C �

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L(a, b) · (c, d) = (ac− bd, ad+ bc)

(c, d) · (a, b) = (ca− db, da+ cb)

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abR[h?o�R ��`ChC\pn�����n�� \C�]�.R[h<¸

R ⊂ CM�P1\

(a, 0)+(b, 0) = (a+b, 0)n?hGk

(a, 0)·(b, 0) = (ab, 0)¸N�ZiZQOa�abR[h

k�QSR�m�`Ca�Y?qSR�^?R!n?hGk�X.R�R[qOqSRW��k?k�QOiZQS`Ch�n?hGk�°n?qOiZQOY?qOQOmC\piZQS`Ch ?� R[X.R[QOh<MFP1\(a, 0) · (c, d) = (ac, ad)

¸j� R[��QOXZm_ik�QSR �n?qOiZQOY?qOQOmN\piZQS`Ch R[QOhGR[X�m�`Ca�Y?qSR�^?R�r/\pt?q a�QOi�R[QOhGR[X�X.R�R[qOqSR[h r/\pt?q

aR[QOhGR�� ��� $�� #V�� ��k?R[X�m�`Ca�Y?qSR�^�R[h

r/\pt?q<n?a k?R[h��\pmFi.`CXaMG© R[abR[XZm�R1�¤R[QOi.R[X@¸kG\C�.�

i2 = i · i = (0, 1) · (0, 1) = (−1, 0) = −1.

��qS�.`�QS�Ziii]\pi.��Cl]t?qOQSl.tsR[QOhGR�½�n&\Ck�X]\pi���n?XZ�@R[q�\pnG� −1

M?®�R[QOi.R[X!QS�.i

z = (a, b) = (a, 0) + (0, b) = (a, 0) + (b, 0) · (0, 1) = a+ b · i,

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� ��5 6�798 � =GB+;Â8_EJ8FHI8FED8 5&Ã

�.` kG\C�.�)��QOX�n?hG�.R[X.R!\pqOi.R!¼_l]t?X.R[QO�j�¤R[QS�]RW��n?X Gl]mj¡GhGk?R[h<MN® QOX��¤R[X.k?R[h�\p� �ZR[iZ��iIk�QSR!¼_l.t?X.R[QO�F�¤R[QS�.Rz = (a, b)h?QSl.tFiWabR[t?X!�/R[hjn?iZ�@R[h<¸�.`ChGk?R[XZh�hFn?X!hG`Fl]t�k�QSR�¼_l]t?X.R[QO�j�¤R[QS�]R

z = a+ b · i.

® QSR ��QOX�o�R��.R[tGR[h�t&\p� R[h<¸��¤R[X.k?R[hbQOhbk�QSR��.R[X ��`Ci]\piZQS`Ch�k�QSR . ���"������!"1n?hGk 2 � �����M5#��� #&������!"U�.R[qO�G�.��R[X.�Zi�phGk_�qOQSl.t<M

6y8�E+7 4 7 0 E � �O³ £ �Ziz = a + b · i ∈ C

R[QOhGR�m�`Ca�Y?qSR�^�Rbr/\pt?q ¸Ä�.`{tGR[Q�NiRe(z) := a ∈ R

k?R[X�� $���� � $�� �n?hGk

Im(z) := b ∈ Rk?R[X � % ���"� �"��� $ � �?��`Ch

zMÄr/\pt?qSR[hsk?R[X �G`CXZa

z = a + 0 · i = a�ZQOhGk¾k�QSRUX.R�R[qOqSR[h

r/\pt?qSR[h<¸Dr/\pt?qSR[h�k?R[Xc�G`CXZaz = 0 + b · i = b · i ¸Ä\pqS�.`�a�QOi Re(z) = 0

¸ � R[X.k?R[h \pnGl.t � $�� �F% ���"� �"� $

% �&�'�($�{o�R[h&\ph?hji@M

6y8�E+7 4 7 0 E � ��¶ £ �Ziz = a+b·i ∈ C

¸@�.`�hGR[h?hFiJa�\phz := a−b·i ∈ C

k�QSR�#&!(% 5 �($ �'$ � !���$� �"� $�� � $<��`ChzM

gI��oNQOqOiz · z = (a+ b · i) · (a− b · i) = a2 + b2 ∈ R.

®�R[h?hz = a+ b · i 6= 0 = 0 + 0 · i ¸GQS�ZiW\pqS�.` z · z > 0

n?hGk�k�QSR�r/\pt?q

w =z

a2 + b2=

a

a2 + b2− b

a2 + b2· i.

t&\piWk�QSR1g�QOo�R[hG�]l.t���i@¸&kG\C�.�z · w = 1

¸?kG\C�.��tGR[QS�]�Zi@¸wQS�.i!k�QSRc£�hF��R[X.�.R1��n

zM

! 8F79> } 798F;98 � �OK

LNM1

3 + 4 · i = (3 + 4 · i)−1 =3− 4 · i

(3 + 4 · i)(3 − 4 · i) =3

25− 4

25· i

e�M(1 + i)(1 + i) = 1− 1 + 2 · i ¸?�.`�kG\C�]�

(1 + i√

2)2 = i.

��qS�.`�QS�Zi 1+i√2

R[QOhGRU½�n&\Ck�X]\pi ��n?XZ�@R[qI\pnG�iM

w�Mω := −1+

√3i

2

M?��R�l.t?hGR[h¾¼FQSRch&\Cl.t<¸GkG\C�.�ω2 + ω + 1 = 0

n?hGkω3 = 1

oNQOqOi@M

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R[QOhGR�m�`Ca�Y?qSR�^�Ryr/\pt?qz = (a, b)

\pqS�R[QOhGR[h � $$# � !"��QOhºk?R[X1g�� R[hGR�\pn�<\C�.�Zi.RNM2s\phs�.Y?XZQSl.tjiU\pnGl]tº��`Chºk?R[X e �2`���() ��� $� % �&�'�($�#$+! $ #$1`_k?R[X\pnGl.t e �2`8W�W� MON � �D��U/��g��/R[hGRNM© R[QDk?R[X � !"��"�=�'�"�L) � $�� � �� ��� R��]l.t?X.R[QO�/R[h°��QOX�k�QSRU��n?h?m_i.R�k?R[X1r/\pt?qSR[hGR[� R[hGR�h?QSl.tFi a�QOi��WR@\pq � n?hGk�£ a�\��oNQOh��pXZi.R[QOq ¸G�.`ChGk?R[XZhya�QOi!k?R[a ���G�Zi]\phGk

r��n?a ÇWX.�ZY?XZn?h?o�n?hGkyk?R[aÈ® QOh?m�R[q��

ϕa�QOi!k?R[X�X.R�R[qOqSR[h���l.tG�.R@L

z = r · (cos(ϕ) + i · sin(ϕ))r

��QSR[XZ� R[QÄQS�Zi

|z| :=√

a2 + b2 = rk?R[X��F. !$) !"� � ��� ! $ �������y��`Ch

z,

Arg(z) := arctan(b/a) = ϕkG\C� . � � �*% $� � ��`Ch

z.

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k?R[X!t?QSR[XWk?R�¡Gh?QSR[XZi.RU© R[iZX]\po{k?R[a �.l.tG`Ch�� R[mN\ph?hji.R[h<M

g�X.�Zi�a�QOi ��QOq��Rbk?R[X��D`Cq�\pX.kG\pX.�.i.R[qOqOn?h?o¾��QOX.k k�QSR�o�R�`CabR[iZXZQS�.l]tGR�© R�k?R[n?iZn?h?o¾k?R[X °n?qOiZQOY?qOQOmC\piZQS`Ch ��VCqOqOQOom_q�\pX�L�© R[iZX]\Cl.tji.R[h���QOXIm�`Ca�Y?qSR�^?R�r/\pt?qSR[h

z = |z|(cos(ϕ)+ i · sin(ϕ))n?hGk

w = |w|(cos(ψ)+ i · sin(ψ))M

��� :BJ$Y���P�� : IX@�� I?O JLY���P ∗ �!�ZR �VP † �!��"�"# V"J I�KHEDI�<M:$�+I�< X 9�% J &;SXS=P ∗ � ��� � P † �$��Q�Q'�( S O$<)�V@+*�,�:=<3T : I-�DI�:=<?@�: <MY � :=K��0J > : A�% I�J � &;Y�� . P�O :OY;K)/$Y O :�U ( <6Y <MY � :=K3T P$Y 90 0 :=YDSW@ �DI�< X 9 @ :=<MY;:OI1.�P�O$:=Y;K)/$Y O :T P$Y π/2 U <?I�,": I2�D:=Y���:O<6SW@�:OY;S���<?@3�D:OI3. P�O :OY;K)/$Y O : JLI-� :=<?@�:=Y U

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� � � ��0 ;9= 2 ~ = 2 > 4 8_;9; Á EJ´ : 0� } ;98j�<8 2y� =GB+;Â8_E 5��

P1\ph?h�QS�Zi

z · w = |z||w|(cos(ϕ) + i · sin(ϕ)) · (cos(ψ) + i · sin(ψ))

= |z||w|(cos(ϕ) cos(ψ)− sin(ϕ) sin(ψ)

+i · (cos(ϕ) sin(ψ) + sin(ϕ) cos(ψ))).

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n?hGksin

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cos(ϕ+ ψ) = cos(ϕ) cos(ψ)− sin(ϕ) sin(ψ),

sin(ϕ+ ψ) = cos(ϕ) sin(ψ) + sin(ϕ) cos(ψ),

�Zi.R[tFi!t?QSR[X!h?QSl.tFi.��\phGk?R[X.R���\pqS�

z · w = |z||w|(cos(ϕ+ ψ) + i · sin(ϕ+ ψ)).

® QOX�t&\p� R[h°\pqS�]` ��`CqOo�R[hGk?R[h°��QSl.tjiZQOo�R[hº¼�\piZ�c�/R[��QSR��.R[h�L

= 4 µ � �9Ã�� © R[Q?k?R[X " Á ; 4 7 } ;�79:�= 4 7 0 E ��`Ch�m�`Ca�Y?qSR�^?R[h{r/\pt?qSR[h��¤R[X.k?R[h{k�QSR ! 8 4@2�� ´8 �sÁ ; 4 7 } ;�79µj798 2�4n?hGk�k�QSR ��2 ´ Á+� 8_E 4 8°= ~+~ 798 2�4 M � 31,&�(�F ( : < < ( &�� ��� & ��F�&�( & : � : / � &�5 +L& mod 2π

/N&�5�����

&�5� < @ $ @ 1#(H"E 7 / 5 � � $N>*( /N&�� ( &�> 7 � $'& � 5 2π � @

|z · w| = |z| · |w|

Arg(z · w) = Arg(z) +Arg(w) mod 2π

! 8F79> } 798F;98 � ��� LNM!P�QSR �WR[o�R[q�2 � ��*) % �"�&2 � ���) � )�� � �-�*)�¸(−1)·(−1) = 1

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Arg(−1) = πM °n?qOiZQOY?qOQOmC\piZQS`Ch�m�`Ca�a�i

��k?k�QOiZQS`Ch�k?R[XD��XZoNn?abR[hji.R¤oNqSR[QSl.t<¸�n?hGkcQOh�k?R[XD·+\piDQS�ZiArg((−1)·(−1)) = Arg(−1)+Arg(−1) =

2π = 0 mod 2πM

e�M!P1\Arg(i) = π/2

¸pQS�Zi)hFn?h�\pnGl]t�mFq�\pX@¸p� \pXZn?ai2 = −1

MC© R[abR[XZm�R@L-°n?qOiZQOY?qOQOmC\piZQS`Ch�a�QOii�/R[��QOXZmFi

R[QOhGR90 � ��P�X.R[tFn?h?o�o�R[o�R[h¾k�QSR1ÇWt?X@M

w�M�¼_R[Qz := 1+i√

2

M?gI��QS�.i |z| = 1n?hGk

Arg(z) = π/4P R��Z� R[o�R[h�QS�Zi

z2 = iM

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£ h��+`Cq�\pX.kG\pX.�Zi.R[qOqOn?h?oVL

Arg(w) =π

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2

|z| =

0 + (3

2)2 =

3

2

��qS�.` Lw · z =

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2 · (cos(π4 + 3π2 ) + i · sin(π4 + 3π

2 )) = 3 ·√

2 · (cos( 7π4 ) + i · sin( 7π

4 ))

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(1 + i)(2 + i)(3 + i) = 10i.

P1\E°n?qOiZQOY?qOQOmN\piZQS`Ch ��`Ch�m�`Ca�Y?qSR�^?R[h r/\pt?qSR[h k�QSR���k?k�QOiZQS`Ch k?R[XU��XZoNn?abR[hji.R{� R[��QOXZm_i@¸Ä�/R�k?R[n?i.R[ik�QSR���¸GkG\C�.�!k�QSRU¼Fn?a�abR�kG\C�!�WXZoNn?abR[hFiW��`Ch

i¸GkÄM t<M

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� ��� ����� � �2����� �?� ��� ����� ��� �P�QSRUr/\pt?qSR[h°k?R[X!�G`CXZa

cos(ϕ) + i sin(ϕ)

�ZQOhGk�o�R[h&\pn�k�QSR�r/\pt?qSR[h°\pn��Ik?R[a � �#"$����F)L# � $�� )

S1 := {z ∈ C | |z| = 1}.

����

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© R[abR[XZm�RN¸GkG\C�.�z, w ∈ S1 ⇒ z · w ∈ S1, z−1 ∈ S1 M

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C \ {0} � R[� ?oNqOQSl]t�k?R[X °n?qOiZQOY?qOQOmC\piZQS`Ch<M_® QOX¦� R[X.k?R[hyn?hG�¦QOa�ZY��pi.R[X�\pnG� � ?t?XZqOQSl]tGR[XWa�QOi���XZn?Y?Y/R[h°\pnG�.R[QOh&\phGk?R[X.�.R[iZ�@R[h<MP1\ °n?qOiZQOY?qOQOmC\piZQS`Ch{��`Ch�m�`Ca�Y?qSR�^?R[h�r/\pt?qSR[h{k?R[X¤��k?k�QOiZQS`Ch�QOt?X.R[X¤��XZoNn?abR[hji.R�oNqSR[QSl.t?m�`Ca�a�i@¸_QS�Zi¦R���m_q�\pX@¸kG\C�.�

(cos(ϕ) + i sin(ϕ))n = cos(nϕ) + i sin(nϕ).

P�QSR��!QS�Zi�k�QSR g � � � R h N �����]l.tGRU�&`CXZabR[q M&TW`Ca��?QOh?QSR[XZi a�QOi�k?R[a ��R[��i.`ChG�.l.tGR[hs© QOhG`Ca�QOn?a R[XZoNQO�?i�k�QSR���G`CXZabR[qOh � ?X

cos(nϕ)n?hGk

sin(nϕ)\pqS�W�D`Cq��_hG`CabRcQOh

cos(ϕ)n?hGk

sin(ϕ)M

! 8F79> } 798F; � � � � ?Xn = 3

¡GhGk?R[h���QOX

cos(3ϕ) + i sin(3ϕ)

= (cos(ϕ) + i sin(ϕ))3

= cos3(ϕ) + 3i cos2(ϕ) sin(ϕ) + 3i2 sin2(ϕ) cos(ϕ) + i3 sin3(ϕ)

= cos3(ϕ)− 3 sin2(ϕ) cos(ϕ) + i(3 cos2(ϕ) sin(ϕ)− sin3(ϕ)).

��qS�.`�QS�Zi

cos(3ϕ) = cos3(ϕ) − 3 sin2(ϕ) cos(ϕ), sin(3ϕ) = 3 cos2(ϕ) sin(ϕ)− sin3(ϕ).

� �3�DI�JL9(J�� �D:��BP < T I�: P ∗ �WRZR �+P † � �ZQ �

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tGR[Q�Nin �

� $ � � $ � �F)�� � ��� $�� ¸��¤R[h?h

zn − 1 = 0

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1 = |zn| = |z|n ¸GkÄM t<M |z| = 1M

= 4 µ � ��5&5 ��"%/�(213+�/N&�5 n n �

+L& ��(*5N$'&�(?+ " F : �;&�>*5 @ ��" "W(*5 < < ( &�� � +L&�5 �;&�5

ζk, k = 0, 1, 2, . . . , n− 1,

F�1,&�(

ζ = ζn := cos(2π

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n).

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&�(*53/N&." �L$ : ( & 1N&�5 (H" + @DE&GF�&�(H" @ ¼_l]t?X.R[QO� R[hº��QOXc\pqS�.` z = cos(ϕ) + i sin(ϕ)

¸ �]`{QS�.izR[QOhGR

n� i.R�gIQOh?tGR[QOi.�Z��n?XZ�@R[q�o�R[h&\pn�kG\ph?h<¸

�¤R[h?h1 = zn = (cos(ϕ) + i sin(ϕ))n = cos(nϕ) + i sin(nϕ).

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cos(nϕ) = 1, sin(nϕ) = 0.

P1\C��tGR[Q�Ni�Lnϕ = 0 mod 2π, nϕ = k · 2π � ?X!R[QOh

k ∈ Z.

®�R[h?h���QOXϕ ∈ [0, 2π)

� �pt?qSR[h°�¤`CqOqSR[h<¸R[XZt&\pqOi.R[h���QOX!k�QSRc�<VN�.n?h?o�R[h

ϕ =k

n2π, k = 0, 1, 2, . . . , n− 1.

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n n� i.R1g�QOh?tGR[QOi.�Z��n?XZ�@R[qOh<¸Gh��pa�qOQSl]t°k�QSRUr/\pt?qSR[h

cos(2πk

n) + i sin(

2πk

n)

P1\�a�QOiWk?R[X0UD� � � R h N ���.l]tGR[h°�G`CXZabR[q

cos(2πk

n) + i sin(

2πk

n) = (cos(

n) + i sin(

n))k

oNQOqOi@¸G�ZQOhGkyk�QSRUg�QOh?tGR[QOi.�Z��n?XZ�@R[qS� \pqS�.`�o�R[h&\pn�k�QSRc�+`Ci.R[h?�@R[h

1 = ζ0, ζ, ζ2, . . . , ζk, . . . , ζn−1.

½�gIP

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� � � 6�798 � 7�EDBD8F7 4 > 3�ÁD2 µ�8F;�E � �

! 8F79> } 798F;98 � �95 �

1QS�Zi!k�QSR1R[QOh?��QOo�R�R[X.�Zi.R1g�QOh?tGR[QOi.�Z��n?XZ�@R[q M

1n?hGk −1

�ZQOhGkyk�QSRc���¤R[Q<��� R[QOi.R[h°g�QOh?tGR[QOi.�Z��n?XZ�@R[qOh<M

1¸ω := −1

2 +√

32 i

n?hGkω2 = −1

2 −√

32 i

�.QOhGk�k�QSR1k�X.R[QJk�XZQOiZi.R[h°gIQOh?tGR[QOi.�Z��n?XZ�@R[qOh<MP�QSR �FQSR[XZi.R[h°g�QOh?tGR[QOi.�Z��n?XZ�@R[qOhs�ZQOhGk

1, i,−1,−i, 1© R[abR[XZm�RN¸FkG\&��R[QOh

n� i.R�g�QOh?tGR[QOi.�Z��n?XZ�@R[q<\pnGl.t�R[QOhGR

2n� i.RN¸

3n� i.R!n?hGk{\pqOqOo�R[abR[QOhyR[QOhGR

k ·n � i.RWgIQOh?tGR[QOi.�#���n?XZ�@R[q<QS�Zi@M

n=1 n=2

n=5 n=6

n=3

n=4

P�QSR°¼Fn?a�abR�k?R[Xn� i.R[h gIQOh?tGR[QOi.�Z��n?XZ�@R[qOh¯mN\ph?h a�QOi���QOq��R�k?R[X��G`CXZabR[q � ?X�k�QSRyo�R�`CabR[iZXZQS�.l]tGR���R[QOtGR

o�R ��n?hGk?R[h��¤R[X.k?R[h�L1 + ζ + ζ2 + . . .+ ζn−1 =

1− ζn1− ζ = 0

= 4 µ � ��5 � ��&�(w 6= 0

&�(*5'& 4��� �

> &� �&��� $N> @ �! 5N5 /�(213+E&."E/N&�5 n#& : " � $N( & < &�5'&�� �G"�;53/N&�5 z ∈ C

< & :�� > &�(��L$ ;53/

zn = w.

�B( &." & 1#(*> < &�5 < ( & � � 4 � ;5N4�+L& &�(*5'&." : &W/N&�> ��� )( /N&�5 n � � �L4 " @DE&GF�&�(H" @ ¼_l.t?X.R[QO�/R w = r(cos(ϕ)+ i sin(ϕ))

MjP�QSR�r/\pt?qz0 := n

√r(cos(ϕ/n)+ i cos(ϕ/n)

R[X�� ?qOqOi¤` ÄR[h���ZQSl]tjiZqOQSl.t�k�QSR�© R�k�QOh?oNn?h?o

zn0 = wMFgIX�� ?qOqOi

z1R[�/R[h�� \pqOqS�

zn1 = w¸_�.` � `CqOoNi k�n?X.l.ty·JR[QOqOn?h?o

(z1/z0)n = 1

¸kÄM t<M

z1/z0QS�Zi!R[QOhGR

n� i.R1g�QOh?tGR[QOi.�.��n?XZ�@R[q M��qS�.`�QS�Zi

z1 = ζ · z0� ?XWR[QOhGR

n� i.RcgIQOh?tGR[QOi.�Z��n?XZ�@R[q

ζM ½�gIP

� =?>@ApBD8FE 2 8jACBDED8 2 =G;9´ 0&2 7 4 B �sÁ >bµ ÁD2{�¦| > Á EJ´�� 0 Ezn = w �

LNM��1R[o�R[� R[hw = a+ ib ∈ C

M

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�?³ � 6 ��� z���� �U���C6 ��� z�� " �U�I�������§�������¦���

e�M!© R��ZiZQOa�abRr := |w| =

√a2 + b2

n?hGkϕ := Arg(w) = arctan(b/a)

Mg¦�!QS�Zi!kG\ph?h

w = r(cos(ϕ) + i sin(ϕ))

w�M!© R��ZiZQOa�abRn√r¸ϕ/n

¸cos(ϕ/n)

n?hGksin(ϕ/n)

MP�QSR�r/\pt?q

z0 = n√r(cos(ϕ/n) + i sin(ϕ/n))

QS�.i!R[QOhGRc�<VN�Zn?h?oGM

�GM!��qOqSRU\phGk?R[X.Rc�JVN�Zn?h?o�R[h��ZQOhGkζz0,

� `C� R[QζR[QOhGR

n� i.R1g�QOh?tGR[QOi.�Z��n?XZ�@R[qDQS�Zi@M

! 8F79> } 798F; � z3 = 1 + i r = 1.4142 . . .¸ϕ = 0.7856 . . .

3√r = 1.1225 . . .

¸ϕ/3 = 0.2595 . . .

¸cos(ϕ/3) = 0.9569 . . .

¸sin(ϕ/3) = 0.2565 . . .

M

z0 = 1.2245 . . . (0.9569 . . . + 0.2565 . . . i) = 1.08 . . . + 0.29 . . . i

P�QSR ���¤R[Q)\phGk?R[X.R[h��<VN�Zn?h?o�R[h¾�ZQOhGk

z1 = ω.z1 = −0.79 + 0.79i

n?hGkz2 = ωz1 = −0.29− 1.08i.

� ��� �0 �#����� ��� �� � �2�;� ������� ��� � � � �����P�QSR�©�\p� �FqS`Ch?QSR[X��/R��]\&��R[h���`CX R[i � \

4000�j\pt?X.R[h��.l.tG`ChyR[QOhGR0°R[iZtG`_k?R���n?X �JVN�Zn?h?o�k?R[X�� � ����������� ) � � $�

C �($���� ��� �x2 + px+ q = 0.

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x1,2 =−p±

p2 − 4q

2.

��Y YD<6:!�D:OI�K)/$Y �D<MS X�9;:=Y � X 9 &DK6:OY , <?I2� �D<M:� : I�K6:O<M@-&DY#O �V:OI p � q � E#PLI ��:=K Y;< X�9 @ ��:=9DI�� :=9;J$Y �D:=K?@WU� <M:�E#P$I ��:OK�SW@�:O9+@ J &��:=<MY;: I S PZO :=Y;J$YDY+@�:OY������������ ������� �3&DY�� � &;SXS�YD< X 9 @ J &DS ,":=Y �D< O%O :=KM:OI�Y @�,": I2�D:=Y U! : I�"$#&%(')%+*�,-�/.�P Y *O:OI�Y 9;JL@ K K)/LY;: P;SX:=<MY;:E#PLI�S X 9 &;Y O S�J �D@�:=<MK &;Y O :=Y <MYDS �3&;SXK6J$Y � J &;S *!&DK J.O :OI�Y U

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� �O³ �¦| > Á ED´ ~ 8 2 : Á H+79>@ACBD8FE À ;98F79ApB Á ED´ �¶

s\ph�R[XZt��pqOi�k�QSRp�q���G`CXZabR[qDk�n?X.l]t � � ���"������� ) ��� $ � � �" � �� �NM&P \

(x+ 12p)

2 = x2 + px+ 14p

2 ¸�� `CqOoNi

x2 + px+ q =(

x+ 12p)2 − 1

4p2 + q = 0 .

��qS�.`�QS�Zi(

x+ 12p)2

= 14p

2 − qn?hGk��.`Ca�QOi

x = −12p±

14p

2 − qg�QOh�¥IR[X�� \pt?X.R[h<¸?n?a �]`Co�R[h&\ph?hji.R #D� !�� ) � � $ C �$ �����*� � $��k?R[X��G`CXZa

y3 + ay2 + by + c = 0

��n�qSVN�.R[h ��n?X.k?RbQOa16

�F\pt?XZtjn?hGk?R[XZi�QOh £�i]\pqOQSR[h R[hji.k?R�l.m_i@M � � UD�Vb Y � NAN � , � n?hGk � � YI�D��[N�D�/� ,B,C¸\pnGl.t�� �"� � ���"� ���yo�R[h&\ph?hji@¸ � \phGk?R[h n?h&\p�?t��ph?oNQOo ��`Ch R[QOh&\phGk?R[X{R[QOhGR[h �JVN�Zn?h?o��Z�¤R[oG¸¦t?QSR[qOi.R[h QOt?h \p�/R[Xo�R[tGR[QOa�M e � c � N U�D�<� ,2� Y?n?�?qOQO��QSR[XZi.R�·D\pXZi]\poNqOQ�\C� �<VN�Zn?h?o��ZabR[iZtG`_k?RcQOh¾�.R[QOhGR[a © nGl.t . �L) 2 ���" �

�) ��

� $ � $ ��$ �(� � � ) �"� �'$+! ���"��� � )�¸/kG\C�1545

R[X.�.l]t?QSR[h<Mļ_R[QOi.k?R[a �ZY?XZQSl]tji a�\ph°��`Chsk?R[X�� �"�=�'�" � ) ��� $ �� !"� % $�� M� ?X R[QOhGR P1\pX.�Zi.R[qOqOn?h?o�k?R[X k�X]\pa�\piZQS�.l.tGR[h°� R��]l.t?QSl.tFi.RN¸?�ZQSR[tGR�©�M �¦M\hj�D� UD� N �D� N UJ�G�?L?� ��QS�Zi.`CX ��` ���qOo�R[�?X]\�M

P R[X�R[X.�Zi.R�¼_l.t?XZQOiZi�QS�ZiUk�QSR #D� !�� ) � � $ � ��" � � �N¸Jn?a k?R[h �Fn&\Ck�X]\piZQS�.l.tGR[h ·JR[XZa qS`N�Z��nj� R[X.k?R[h<MIrnGR[X.�ZiX.R�l.t?hGR[i�a�\ph�h&\Cl.t<¸&kG\C�.�

(

y + 13a)3

= y3 + ay2 + 13a

2y + 127a

3 .

®�R[h?h���QOX!hFn?hx := y + 1

3a�.R[iZ�@R[h<¸�¡GhGk?R[h���QOX

y3 + ay2 + by + c = x3 − 13a

2(

x− 13a)

− 127a

3 + b(

x− 13a)

+ c .

© R[hjn?iZ�@R[h°��QOX!hjn?h���n?X��W�?m� ?XZ��n?h?o

p := −b+ a2

3 q := 13ab− c− 2

27a3,

�.`�R[XZt&\pqOi.R[h°��QOX!k�QSR���qSR[QSl.tFn?h?ox3 = px+ q.

P�QSR�m_n?�?QS�.l.tGR¤g�XZo��ph?��n?h?o � ?t?XZiD\pqS�.`!h?QSl.tji+k�QOX.R[mFiD��n?XJ�<VN�Zn?h?oG¸��.`ChGk?R[XZh � ?t?XZi<��n?X Gl.mU\pn��GR[QOhGR¦mFn?�?QS�.l]tGR��qSR[QSl]tjn?h?o{`Ct?hGR �jn&\Ck�X]\piZQS�.l]tGR[h°·DR[XZa�M8��� X=<��;< P YD: �D:=K*E;:OIXI?P ∗ ����R Q+P † ��Q "ZR8X8� < X9P K P0E#P$Y+@�JLY(J+P ∗ � � SZSVP † �WQ�Q��8�J % <?I?P$K J �%P V"JLI2�DJ$Y#PVP ∗ �WQZY � P † �WQ �.R

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�GK � 6 ��� z���� �U���C6 ��� z�� " �U�I�������§�������¦���

! 8F79> } 798F;98 � �95_³® QOX�� ?t?X.R[h�\phyk?R[X!� qSR[QSl.tjn?h?o

y3 + 3y2 + 3y+ 9 = 0k�QSR�m_n?�?QS�.l.tGR g�XZo��ph?��n?h?o{k�n?X.l.tyn?hGk�� R[hFn?iZ�@R[h

(y + 1)3 = y3 + 3x2 + 3y + 1¸&\pqS�.`

y3 + 3y2 + 3y + 9 = (y + 1)3 − 3y − 1 + 3y + 9 = (y + 1)3 + 8 = 0.

°QOix = y + 1

��QOX.k�k�QSR���qSR[QSl]tjn?h?o���nx3 = −8,

� \C�x = −2

\pqS���<VN�Zn?h?o�qOQSR � R[XZi@M+¼_`Ca�QOicQS�.iy = x − 1 = −3

R[QOhGR��<VN�.n?h?o�n?hG�.R[X.R[XUn?X.�.Y?X ?h?oNqOQSl.tGR[h��qSR[QSl]tjn?h?oGMP�QSR���qSR[QSl]tjn?h?o

y3 +3y2 +9 = 0¸_k�QSR $ � +� � ��� $�� � ��)$) � $ � �/\pqS�¤kG\C�¤qSR[iZ��i.R�© R[QS�ZY?QSR[q ¸!� ?t?XZi¦a�QOi

x = y+1��n�k?R[X���qSR[QSl]tjn?h?ox3 − 3x+ 11 = 0 .

® QOX!�.l.tGR[QOhGR[hs\pqS�.`�h?QSl.tFiW�_QSR[qÄo�R[�¤`Ch?hGR[h¾��nyt&\p� R[h<¸G`_k?R[X!k?`_l.t±s\ph �/R[hGVCiZQOoNi �ZR[iZ��i�R[QOhGR{hGR[nGR{£ k?R�RN¸<n?a �¤R[QOi.R[XZ��n?m�`Ca�abR[h<MIP R[X��&\pqOq

p = 0QS�.iUh&\pi ?XZqOQSl.t R[QOh�� \Cl.t<¸

n?hGk�kG\pa�QOi�k� ?X���R[h���QOX�\ph?hGR[t?abR[h<¸&kG\C�]�p 6= 0

MGP�QSR��!��QOX.k��.QSl.t�QOh°T� ?XZ�@R�\pqS�!�/R��FnGR[a tGR[X]\pnG�.�Zi.R[qOqSR[h<MP�QSR ��R[XZ�?q� �/R[hGk?R[h¾£ k?R�RcQS�Zi!R��@¸

x = u+ v

��n��.R[iZ�@R[h<M® QOX�X.R�l.t?hGR[h�h&\Cl]t�L

x3 = u3 + 3u2v + 3uv2 + v3)

= 3uv(u + v) + u3 + v3

= 3uvx+ u3 + v3

��hGk?R[X.R[X.�.R[QOi.�x3 = px+ q .

gI��X.R[QSl]tji�kG\pa�QOi�\pnG�@¸�h&\Cl]t���`CqOo�R[hGk?R��1¼ �_�.i.R[aÈ��`Ch¾��qSR[QSl.tFn?h?o�R[h°h&\Cl]tun?hGk

v��nyqSVN�.R[h<M

3uv = p

u3 + v3 = q

P1\���QOXb\ph?o�R[hG`Ca�abR[h t&\p�/R[h<¸+kG\C�.�p 6= 0

¸+t&\p� R[h ��QOXu 6= 0

M ��ÇWhGk h&\pi ?XZqOQSl.t \pnGl]tv 6= 0

M ����nG�k?R[X R[X.�Zi.R[hº� qSR[QSl.tjn?h?o#��`CqOoNi kG\ph?h

v = p3u

¸ �.`{kG\C�]�W��QOXv\pnG��k?R[X����¤R[QOi.R[h�R[qOQOa�QOh?QSR[X.R[hsm�VCh?hGR[h<M ® QOX

�/R[m�`Ca�abR[hu3 +

( p

3u

)3= q .

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� �O³ �¦| > Á ED´ ~ 8 2 : Á H+79>@ACBD8FE À ;98F79ApB Á ED´ �/Ã

°n?qOiZQOY?qOQO��QSR[X.R[h�a�QOiu3 qOQSR � R[XZi@M

(

u3)2 − qu3 +

p3

27= 0 .

P�QSR���QS�Zi!R[QOhGR �Fn&\Ck�X]\piZQS�.l]tGR���qSR[QSl]tjn?h?o�QOhu3 � P�QSR1�<VN�Zn?h?obQS�.i!�.`Ca�QOi

u3 =q

2± 1

2

q2 − 4(p

3

)3=q

2±√

(q

2

)2−(p

3

)3.

® QOX�hGR[t?abR[h°k�QSR+�<VN�Zn?h?oVL , �

u =3

q

2+

(q

2

)2−(p

3

)3.

P1\v = p

3u

¸�� R[m�`Ca�abR[h°��QOX�QOhG�Zo�R��]\pa�i�k�QSR1�<VN�Zn?h?ox = u+ v

M

= 4 µ � ��5�¶ ��� : < 5N(H" � $'&��� :�� &�> � � 8;: p 6= 0

$ + < ( & � > &�(�� $ ;53/ x3 = px+ q < ( & � �G"�;53/

x = u+p

3u,

F�1,&�(

u =3

q

2+

(q

2

)2−(p

3

)3.

! 8F79> } 798F;98 � �95�K

LNMx3 = 9x+ 28

M��QSR[X!QS�ZiW\pqS�]`

p = 9n?hGk

q = 28M&®¯QOX��/R[m�`Ca�abR[h � ?X

uL

u =3

14 +√

142 − 33

=3

14 +√

169

=3√

27

= 3 .

P1\pa�QOi

x = u+p

3u= 3 +

9

3 · 3= 3 + 1 = 4 .

P�QSR��.R���qSR[QSl.tFn?h?o�t&\pi!QOh�k?R[XW·+\pi!hjn?X�R[QOhGR1X.R�R[qOqSRc�<VN�Zn?h?oG¸�ZQSR[tGR ��?n?h?o�R[h<M

8 ��� Y��V:=Y��3&�� O J � :=Y � G;SXSX:OY��+<6:1*O:=< O :=Y�P �;JLSXS3�D<M: − JL9;K *!& �V:OI O KM:=< X�9;:=Y E P$I ��:=K&� GD9DIX@WU

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��� � 6 ��� z���� �U���C6 ��� z�� " �U�I�������§�������¦���

e�Mx3 = −3x+ 4

M��n?h�QS�Zi

p = −3¸q = 4

n?hGk���QOX�� R[m�`Ca�abR[h¾k�QSR1�<VN�Zn?h?o

x = u+p

3u= u− 1

u,

a�QOiu =

3

2 +√

5 .

°QOiWR[QOhGR[a ·+\C�.l.tGR[h?X.R�l]t?hGR[XW¡GhGk?R[i�a�\ph

u ≈ 1, 61803

1

u≈ 0, 61803 ,

�]`�kG\C�]�x ≈ 1, 0000

M<® QSR�a�\phºk�n?X.l]t�g�QOhG�.R[iZ�@R[h � R��Zi.�Zi.R[qOqOi@¸ QS�Zix = 1

R[QOhGR��<VN�Zn?h?oGMJÆ ÄR[hj�&\pXQS�.i

1 =3

2 +√

5− 13√

2 +√

5!

® QOX�� R[iZX]\Cl]tji.R[h¾hjn?h�kG\C�!© R[QS�ZY?QSR[qx3 = 6x− 4 .

P�n?X.l.t�R[QOhGR¤T n?XZ��R[hGk�QS�.mFnG�.�.QS`Ch���`Chf(x) := x3−6x+4

m�VCh?hGR[h���QOX�� R��Zi.�Zi.R[qOqSR[h<¸�kG\C�.�DR��+k�X.R[Q��<VN�Zn?h?o�R[hoNQO�?i@¸?� `@��`Ch���QOX�R[QOhGRck�QOX.R[mFi!�]R[tGR[h�m�VCh?hGR[h<¸?h��pa�qOQSl.t

x = 2M?P�QSR � \pX.kG\ph?QS�.l]tGRU�G`CXZabR[q+R[XZoNQO�?i

x = u+2

u,

a�QOiu =

3

2 +√−4.

Æ��j� `Ct?qck�QSR°�JVN�Zn?h?o�R[hu\pqOqSR�X.R�R[qOq �ZQOhGkĸ¤�¤R[X.k?R[h ��QOX�\ph k�QSR��.R[X�¼Fi.R[qOqSR¾o�R[����n?h?o�R[h<¸¤a�QOi�m�`Ca�Y?qSR�^?R[hr/\pt?qSR[h ��n X.R�l.t?hGR[h<M gI���Zi.R[qOqOi{�ZQSl.t¯tGR[X]\pnG�@¸¦kG\C�.���Wt?h?qOQSl]tGR���QOa�abR[X{Y&\C�.�ZQSR[XZi@¸I� R[h?h¯k�QSR°� qSR[QSl.tjn?h?ox3 = px + q

k�X.R[Q �9X.R�R[qOqSR��U®ºn?XZ�@R[qOh t&\pi@M�rn c � N U&�+�<�<��r R[QOi���n?X.k?R�k�QSR��]R[X��\pqOq ¸�k?R[X��]`Co�R[h&\ph?hji.R� l@\C�ZnG��QOXZX.R�k�n?��QO�?QOqSR����]¸Ih?QSl.tji�� R[t&\phGk?R[qOi@M c � N U�D�<�«�.R[qO�G�Zi�R�^�Y R[XZQOabR[hFiZQSR[XZi.R��.l]tG`Ch kG\pa�QOi@¸)®ºn?XZ�@R[qOh\pnG��hGR[oj\piZQO��R[h�r/\pt?qSR[h���nb��QSR[tGR[h<¸�`C�j� `Ct?qR[XIR��)��n�k�QSR��]R[X¤r R[QOi?� \p�&\pXZiZQOo"��� \phGkÄM�Ç�hGk�`C�j� `Ct?q�.l]tG`Chb�.R[QOik?R[a °QOiZi.R[q�\pqOi.R[X¤o�R[qSR[o�R[hFiZqOQSl.t�a�QOi¦m�`Ca�Y?qSR�^?R[h�r/\pt?qSR[h{o�R[X.R�l.t?hGR[i ��n?X.k?RN¸��Zi]\phGk?R[h{k�QSR��]R�r/\pt?qSR[h{k�n?X.l]t�.l]tGR[QOhj�&\pX.R�® QSk?R[X.�ZY?X Gl.tGR��?QS����n?a ��h�� \ph?o�k?R���L@v�M��j\pt?XZtFn?hGk?R[XZi.�1QOh¾¥IR[XZXZn��#M ÇWhG�]R[X.R�g�X�� \pt?XZn?h?o�QS�ZiR��@¸_kG\C�.� tGR[n?iZ��n?i]\po�R m�R[QOh�¼FiZnGk�QSR[X.R[hGk?R[X@¸&\pnGl.tyh?QSl]tjiWk?R[X���t �_�.QOm ¸�© QS`CqS`CoNQSRN¸��¤tGR[a�QSR1`Fk?R[X�£ h���`CXZa�\piZQOm/¸R[XZhG�ZiZt&\ ��i.Rc��X.`C�?qSR[abRUa�QOi!m�`Ca�Y?qSR�^�R[hsr/\pt?qSR[h�t&\pi@M

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� �O³ �¦| > Á ED´ ~ 8 2 : Á H+79>@ACBD8FE À ;98F79ApB Á ED´ �$�

® QOX�a G�.�.R[h¾\pqS�.`3√−2 + 2i

¡GhGk?R[h<¸?`_k?R[X�� R��.�]R[X¤o�R��]\poNi@¸?R[QOhGR : 0� } ;98��<8 r/\pt?qua�QOi

u3 = −2 + 2iM?P1\C� QS�ZiW\p�/R[X m�R[QOh���X.`C�?qSR[a�M

−2+2i

1+i

��nG�Uk?R[a¿©¤QOqSk o�R[tji�tGR[XZ��`CX@¸DkG\&�u = 1 + i

R[QOhGR{�JVN�Zn?h?o�QS�Zi@¸J��QSRba�\ph \pnGl.t �.l.t?hGR[qOq h&\Cl.t?X.R�l]t?hGR[i@MP1\ph?h

v = p3u = 2

1+i = 1− i ¸�n?hGk � ?Xx¡GhGk?R[h���QOX

x = u+ v = (1 + i) + (1− i) = 2 ,

� \C�WQOh�k?R[XW·D\pi�R[QOhGRc�<VN�Zn?h?obQS�.i@M

P�QSRy\phGk?R[X.R[h k�XZQOiZi.R[h ®ºn?XZ�@R[qOh ��`Ch −2 + 2i�.QOhGk �ZR[iZ��i

u = ω(1 + i)n?hGk

u = ω2(1 + i)¸)� `C� R[Q

ω = −12 + 1

2

√3QS�Zi@M

P1\ph?h�R[XZt&\pqOi.R[h���QOXv = 5

ω(2+1) = ω2(2− i) M?�WqS�.`

x = u+ v = ω(1 + i) + ω2(1− i)= (ω + ω2) + i(ω − ω2)

= −1 + i2√

3 = −1−√

3 .

��qS�!k�XZQOiZi.Rc�<VN�Zn?h?o{R[XZt&\pqOi.R[h���QOX

x = ω2(2 + i) + ω(2− i)= −1 +

√3 .

P1\pa�QOi!t&\p�/R[h���QOX!k�QSRck�X.R[Q<X.R�R[qOqSR[h�®ºn?XZ�@R[qOh2,−2±

√3k?R[X ��qSR[QSl.tFn?h?o

x3 = 6x− 4o�R ��n?hGk?R[h<M

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� � � 6 ��� z���� �U���C6 ��� z�� " �U�I�������§�������¦���

� �2� � ���?��� � � � �,� ��� � ����� � ����������� �0 �#����� � �� �� � � �;� ��� � � ��� � � � ����� ���

© R[iZX]\Cl.tFi.R�k�QSR m_n?�?QS�.l.tGR���qSR[QSl.tFn?h?oy3 + ay2 + by + c = 0

MLNM�m_n?�?QS�.l]tGR¦g�XZo��ph?��n?h?oGMpP�n?X.l.t�k�QSR ¼Fn?�G�ZiZQOiZn?iZQS`ChGR[h

x = y+ 13a¸p = −b+ a2

3

¸q = 1

3ab−c− 227a

3

o�R[tFi�k�QSRU��qSR[QSl]tjn?h?o ?� R[X�QOh°k�QSRU��qSR[QSl.tFn?h?o

x3 = px+ q .

e�M!© R[X.R�l]t?hGR q2 +

( q2

)2 −(p

3

)3 ∈ CM

��nG�!k�QSR��.R[X r/\pt?q<a G�.�]R[hy��QOXWR[QOhGRcT n?�?QOm_��n?XZ�@R[qu��QSR[tGR[h<M

w�M�® QOX!t&\p�/R[h°�.l]tG`Ch°R[XZm_q �pXZi!��QSRUk�QSR��Wh��ptGR[XZn?h?o��Z�¤R[QS�]R�k�n?X.l]t�� ?t?XZ�&\pX�QS�Zi�L?�/R[X.R�l.t?hGRUa�QOi�k?R[a ·+\���]l.tGR[h?X.R�l.t?hGR[X © R[iZX]\po

rn?hGk��WXZoNn?abR[hFi

ϕk�QSR��.R[X�r/\pt?qJn?hGk�� R[X.R�l]t?hGRUkG\ph?h

u = 3√r(

cos(ϕ

3

)

+ i sin(ϕ

3

))

.

�GM!gIQOhGRc�<VN�Zn?h?o{QS�Zix = u+ p

3u

M �9® \C�!Y&\C�.�ZQSR[XZi@¸?� R[h?hu = 0

± ���M!P�QSRU\phGk?R[X.R[h°�<VN�Zn?h?o�R[h°�.QOhGk

ωu+ ω2 p

3u

n?hGkω2u+ ω · p

3u.

! 8F79> } 798F; � ��5Gà ®¯QOX��/R[iZX]\Cl.tFi.R[h�hFn?h°k�QSRU��qSR[QSl]tjn?h?o

x3 = 15x+ 4 .

P�n?X.l.ts© R[iZX]\Cl.tji.R[hºk?R�� ��X]\pY?tGR[h���`Chf(x) = x3 − 15x − 4

¡GhGk?R[h°��QOX@¸kG\C�.�WR���k�X.R[QD�<VN�.n?h?o�R[h°oNQO�?i���@M/��n��9oj\p�/R[h���¸ �¤`@��`Ch ��QOXcR[QOhGR�k�QOX.R[mFi1�.R[tGR[hºm�VCh?hGR[h<¸/h��pa�qOQSl]t

x = 4M P�QSR�� \pX.kG\ph?QS�.l]tGR[h �G`CXZabR[qOh

R[XZo�R[�/R[hx = u+

5

u,

a�QOiu =

3

2 +√−121.

® QOX�a G�.�.R[h¾\pqS�.`3√

2 + 11i

¡GhGk?R[h<¸<`_k?R[X1� R��]�.R[X1o�R��]\poNi@¸ÄR[QOhGR�m�`Ca�Y?qSR�^?R{r/\pt?qua�QOi

u3 = 2 + 11i =: aM �Wn?hºoNQOqOi |a| = √125

¸\pqS�.` |u| = 3

√√125 =

√5M?� ?X!kG\C�W��XZoNn?abR[hji

ϕ��`Ch

at&\p�/R[h���QOX

tan(ϕ) = 112

M

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� ��¶ 6y8 2��<Á E ~ = � 8FE 4 =G;9>@= 4 µ ~ 8 2�� ;´8jH 2 = � 5

°QOi �WQOq� R R[QOhGR��!·D\C�.l]tGR[h?X.R�l.t?hGR[X.��¡GhGk?R[h���QOXϕ ≈ 1, 3909428

M � ?X!kG\C�!�WXZoNn?abR[hFi!��`Chu¡GhGk?R[h���QOX�L

Arg(u) =ϕ

3≈ 0, 4634761.

P1\pa�QOi!t&\p�/R[h���QOX!© R[iZX]\pobn?hGk���XZoNn?abR[hjiW��`Chuo�R �9n?hGk?R[h�L

u ≈√

5(cos(0, 4634761) + i sin(0, 4634761) ≈ 2 + i

s\ph¾X.R�l]t?hGR[i�k�QOX.R[m_i h&\Cl.t<¸<kG\C�.�(2 + i)3 = 2 + 11i

¸<\pqS�.`�oNQOqOic�]R[qO�G�ZiU��qSR[QSl.t?tGR[QOi@M<� ?Xv�/R[m�`Ca�abR[h

��QOXv = p

3u = 52+i = 2− i ¸Gn?hGk � ?X

x

x = u+ v = (2 + i) + (2− i) = 4 ,

� \C�WQOh�k?R[XW·D\pi�R[QOhGRc�<VN�Zn?h?obQS�.i@M

P�QSR�\phGk?R[X.R�T n?�?QOm_��n?XZ�@R[qOh ��`Ch2 + 11i

�ZQOhGku = ω(2 + i)

¸Dn?hGku = ω2(2 + i)

MD® QOXU�/R[m�`Ca�abR[hv = 5

ω(2+i) = ω2(2− i) MG��qS�.`

x = u+ v = ω(2 + i) + ω2(2− i)= 2(ω + ω2) + i(ω − ω2)

= −2 + i2√

3 = −2−√

3 .

��qS�!k�XZQOiZi.Rc�<VN�Zn?h?o{R[XZt&\pqOi.R[h���QOX

x = ω2(2 + i) + ω(2− i)= −2 +

√3 .

P1\pa�QOi!t&\p�/R[h���QOX!k�QSRck�X.R[Q<X.R�R[qOqSR[h�®ºn?XZ�@R[qOh4,−2±

√3k?R[X ��qSR[QSl.tFn?h?o

x3 = 15x+ 4o�R ��n?hGk?R[h<M

� ��� � �������������������� ���2�;��� � �� � � �2� ��� ���® QOX�t&\p� R[h�k�QSR�r/\pt?qO� R[X.R[QSl]tGRcQOa�abR[X!abR[t?X�\pnG�.o�R[�¤R[QOi.R[i�L

N ⊂ Z ⊂ Q ⊂ R ⊂ C

��R�k?R[X�g�XZ� R[QOi.R[XZn?h?o��.�.l]t?XZQOiZi���n?X.k?Rs��`CXZo�R[hG`Ca�abR[h<¸�n?a§R[QOhGR¾�/R��ZiZQOa�aUi.Rº��XZi���`Ch«��qSR[QSl.tFn?h?o�R[hu��nqSVN�.R[h<MÄ£ a qSR[iZ��i.R[h ¼_l.t?XZQOiZi1� \pX�k�QSR��1k�QSR�� qSR[QSl.tjn?h?o

x2 + 1 = 0MÄP�QSR ��QOh?��n?h&\pt?abR���`Ch

iR[XZabVCoNqOQSl.tFi

h?QSl.tFi�hjn?XIk�QSR!½�n&\Ck�X]\pi���n?XZ�@R[q/\pnG� −1��n���QSR[tGR[h<¸N�.`ChGk?R[XZh�kG\C�¦rQSR[tGR[h���`Ch�� R[qOQSR[�?QOo�R[hb�+`Ci.R[h?����n?XZ�@R[qOh��

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� � � 6 ��� z���� �U���C6 ��� z�� " �U�I�������§�������¦���

k�QSR�� qSR[QSl.tjn?h?ozn = w

t&\pi�QOhCQOa�abR[X�R[QOhGR��<VN�Zn?h?oG¸J�¤R[h?h

w 6= 0�.`Coj\pXco�R[h&\pn

n¼Fi Gl]m MDP�QSR

p�

q���G`CXZabR[qI�@R[QOoNi@¸<kG\C�.�UR[QOhGR�� R[qOQSR[�?QOo�R �jn&\Ck�X]\piZQS�.l]tGR{��qSR[QSl]tjn?h?o

x2 + px + q = 0���.`Coj\pX�� ?X

pn?hGk

qm�`Ca�Y?qSR�^$�U�JVN�Zn?h?o�R[h � R��ZQOiZ��i@M+P�QSR � \pX.kG\ph?QS�]l.tGRy�G`CXZabR[q �@R[QOoNi@¸)kG\C�.��� R[qOQSR[�?QOo�R��+`Cq��FhG`Ca�oNqSR[QSl]tjn?h?o�R[h��`Ca ��X]\Ck w�QOh

CqSVN�.�&\pX��ZQOhGkÄM+gI��oNQO�?i���� \pX�hG`_l.t R�^_Y?qOQO��QOi.R��<VN�.n?h?o�����`CXZabR[qOh � ?X��+`Cq��_hG`CabR���`Ca

��X]\Ck4¸&� R[qSl.tGR���`Ch � \pX.kG\phG`N�c¼_l]t! ?qSR[X ] � Y � NAN � NAR , �o�R �9n?hGk?R[hs��n?X.k?R[h<¸�� `���� R � R , � n?hGk M j�VbH,2�

�@R[QOoNi.R[h{\p�/R[XIn?a1800

¸jkG\C�.�¦�.`CqSl.tGR!�JVN�Zn?h?o���� `CXZabR[qOh � ?X¤�D`Cq��_hG`Ca�oNqSR[QSl.tFn?h?o�R[hytGVCtGR[X.R[h���X]\Ck?R��Ih?QSl]tjiR�^�QS�ZiZQSR[X.R[h<MGÇ�a �.`�� R[abR[XZm�R[hG�.�¤R[XZi.R[X�QS�Zi!k?R[X�� R[X ?t?a�i.R

= 4 µ � ��5���� �<Á E ~ = � 8_E 4 =G;Â>�= 4 µ ~ 8 2�� ;9´&8FH 2 =�� & < & � > &�(�� $ ;53/

xn + a1xn−1 + · · ·+ an−1x+ an = 0

�(?+ai ∈ C

$ +�(*5 < &." +L&�5," &�(*5'& � �G"�;53/ (*5

C @

® QOX��¤R[X.k?R[h°k�QSR��]R[h���QSl.tFiZQOo�R[hº¼�\piZ�cQOh�k�QSR��]R[X!¥I`CXZqSR��Zn?h?obh?QSl]tjiW�/R[�¤R[QS�]R[h<¸GkG\b\ph&\pq��_iZQS�.l.tGR�`Fk?R[X�i.`CY/`p�qS`CoNQS�.l]tGRW��XZoNn?abR[hji.R!h?QSl]tji¤��R[XZa�QSR�k?R[h��¤R[X.k?R[h{m�VCh?hGR[h<M�P R[hb�¤`Ct?q R[QOh�� \Cl]tG�Zi.R[h�© R[�¤R[QS� � ?t?XZi¦a�\phba�QOi��QOq��R�k?R[X m�`Ca�Y?qSR�^�R[h���h&\pq��_�.QS�@MFP R[X�R[X.�Zi.R R[QOh?QOo�R[XZa�\&��R[h��ZiZX.R[h?o�R © R[�¤R[QS�!��n?X.k?R���`Ch c � Y � e �A`�W�W1QOh�.R[QOhGR[XWP `Cm_i.`CX]\pXZ� R[QOi�L�dCvNv�R[XZ�?X]\Cl.tFi@M/�1\pnG�]��� \pXWkG\pa�\pqS��eNe%�j\pt?X.R�\pqOi@M?®¯QSR ��QOXWhG`Fl]t��.R[tGR[h�� R[X.k?R[h<¸mN\ph?h�a�\phy\pnG�¤k?R[a `C� R[h���`CXZa�n?qOQSR[XZi.R[hy�?n?hGkG\pabR[hji]\pqS�]\piZ�ck?R[X ��qOo�R[�?X]\UR[QOh�R[i�� \C���.i�pXZm�R[X.R��¤g�XZo�R[�?h?QS�tGR[XZqSR[QOi.R[h<M

= 4 µ � ��5���� �<Á E ~ = � 8_E 4 =G;Â>�= 4 µ ~ 8 2�� ;9´&8FH 2 =�� � 4 8 � 8 2 >�7 0 E � ��&�(f(x) = xn+a1x

n−1 + · · ·+an�(?+ai ∈ C @ � 5N5 $ +

f&�(*5'& � 4�+

� : (H"�( & : ;53/ (*5 ��(*5'&� :W7 4�+ � : &�5

f(x) = (x− λ1)r1 · · · (x− λk)rk , ri = 1 ,

F�1,&�(

r1 + · · ·+ rk = n 6 λi 6= λj 798;: i 6= j @

s\ph�hGR[h?hjiλi

��n?qOqS�Zi.R[qOqSRc��`Chfk?R[X�¥�QSR[q� \Cl.t?tGR[QOi

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8 ��� P � P�T+< X9P0E;:OIXI�J I�< P ∗ �WQ�"�" P † �WQZR Q8 � K J.P K P �3&���YD< P ∗ �!�ZR�Q+P † �!��"�"8�# <M:=KMS :=YDI�<�� � � :OK P ∗ �!��Y�" P † �!��"ZS

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� ��¶ 6y8 2��<Á E ~ = � 8FE 4 =G;9>@= 4 µ ~ 8 2�� ;´8jH 2 = ��

−i

i

1−1 2−2

−2i

2i

P�QSR1© QOqSk?R[X!�@R[QOo�R[h°k�QSR ��n?qOqS�Zi.R[qOqSR[h�k?R[XW�+`Cq��_hG`CabR

x5 + 3x+ 10

n?hGkx7

7!+x6

6!+x5

5!+x4

4!+x3

3!+x2

2!+ x+ 1.

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�� $����� $ % ��� �'$��

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QS�ZiIR[QOh�ZY/R[��QSR[qOqSR�� g�qSR[abR[hji@¸Än?hGk

ν : N −→ Nk�QSR��?n?h?m_iZQS`Ch<¸/�¤R[qSl.tGR�kG\C� g�qSR[abR[hFi

n\pn��¤�]R[QOhGR[h � � ���L� !"� �'$��

ν(n)\p�?�?QOqSk?R[i@MG® QOX!�.R[iZ�@R[h

1, 2 := ν(1), 3 := ν(2), . . .

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LNM1QS�Zi�m�R[QOh � \Cl.t�� `CqOo�R[X@¸)kÄM t<M ∀n ∈ N : ν(n) 6= 1

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n 6= m⇒ ν(n) 6= ν(m)��k�QSRc�?n?h?m_iZQS`Ch

νQS�Zi�QOh �ZR[mFiZQO����M

e�M�¼_R[QS ⊂ N

a�QOi

• 1 ∈ S

• n ∈ S ⇒ ν(n) ∈ S

P1\ph?h�QS�.iS = N

M

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� � � � 8j: ÁD2 >�7 �8�6y8 )E+7 4 7 0 EJ8_E � ¶

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LNM�¼_R[Qa ∈ C

¸a 6= 0

M?® QOX!k?R�¡Gh?QSR[X.R[han

X.R[mFn?X.�ZQO��k�n?X.l.t

a0 := 1, an := n · an−1.

e�M�® QOXWk?R�¡Gh?QSR[X.R[hn!X.R[m_n?X.�ZQO��k�n?X.l.t

0! := 1, n! := n · (n− 1)!.

w�M�¼FQOhGka1, a2, . . . , an

r/\pt?qSR[h<¸G�]`��.R[iZ�@R[h���QOXn∑

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ai := a1 + a2 + . . .+ an.

��h&\pqS`Con−1∑

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ai = a3 + a4 + . . .+ an−1,

3∑

j=0

ai+2 = a2 + a3 + a4 + a5.

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i=1

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ak

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i=1

ai := a1 · a2 · . . . · an.

= 4 µ � � � � � 8F: ÁJ2 >�7 0 ED> }+2 7�EDµj7 } ��� " +f(1) 7 &." + /N&�> &W/�+ ;5 < 4� 5N5 f(ν(n))

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f(n) 798;: >*> & n ∈ N&�(*5 < &�N+ ( / 7 &." + /N&�> &W/�+ @

� 8j: ÁD2 >�7 � 8�6y8�ED7 4 7 0 E ~ 8 2{� ~+~ 7 4 7 0 E �� ?X

m ∈ Nk?R�¡Gh?QSR[X.R[h���QOX!X.R[m_n?X.�ZQO��R[QOhGRc���?�?QOqSk�n?h?o

m+ : N −→ N;n 7→ m+ nL

{

m+ ν(n) := ν(m+ n)

m+ 1 := ν(m)

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� K � 6 �[� ��� ���y���I�����y� � �I�{���¦���

¼_`bR[XZt��pqOi!a�\ph3 + 3 = 3 + ν(2)

= ν(3 + 2)

= ν(3 + ν(1))

= ν(ν(3 + 1))

= ν(ν(ν(3)))

= ν(ν(4))

= ν(5)

= 6

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m ∈ Nk?R�¡Gh?QSR[X.R[h���QOX!X.R[m_n?X.�ZQO��R[QOhGRc���?�?QOqSk�n?h?o

m· : N −→ N;n 7→ m · n L{

m · ν(n) := m+m · nm · 1 := m

® \C��QS�ZiW\pqS�]`3 · 3 ±

� 79H 0 ED=?A�Aj7�� � =GBD;98_E® QOX¦m�VCh?hGR[h�n?hG� \pnGl]t{m�`Ca�Y?qOQO��QSR[XZi.R[X.R�X.R[m_n?X.�ZQO��RWP R�¡Gh?QOiZQS`ChGR[h���`CX.�Zi.R[qOqSR[h<M_P�QSRW� R[X ?t?a�i.R[h�Y R j �D�/� P P2R , � � r/\pt?qSR[h F (n)

� R[QS�ZY?QSR[qS�.�¤R[QS�.Rc�.QOhGkyk?R�¡Gh?QSR[XZi�k�n?X.l.t

F (1) := 1,

F (2) := 1,

F (ν(ν(n))) := F (n) + F (ν(n)).

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• A(1)(H" +%F $ :

• A(n)⇒ A(n+ 1)

�! 5N5 (H" +A(n)

F $ : 798;: >*> & n ∈ N @DE&GF�&�(H" @ ¼_R[Q S := {n ∈ N | A(n)

QS�.i!� \pt?X } MÄP1\ A(1)� \pt?XcQS�Zi@¸/QS�Zi

1 ∈ S MÄ£ �Zi n ∈ S ¸ �]`�QS�Zi A(n)� \pt?XWn?hGky�.`Ca�QOiW\pnGl.tA(n+ 1)

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S = NM

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� \pt?X�\ph?o�R[hG`Ca�abR[h<M

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! 8F79> } 798F;98 � � ³

• ¼_l]t&\pnGR[h���QOX!n?hG� � `CqOo�R[hGk?R���\ph�L

1 = 1 = 12

1 + 3 = 4 = 22

1 + 3 + 5 = 9 = 32

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1 + 3 + 5 + 7 = 16 = 42,

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1 + 3 + 5 + . . .+ 2n− 1 = n2.

P�QSR���m�VCh?hGR[h°��QOX�a�QOi!��`CqOqS�Zi�phGk�QOo�R[X�£�hGk�n?m_iZQS`Ch��/R[�¤R[QS�.R[h�L

� � E ~ Á : 4 7 0 ED>@=GE ��=GED´ L�P�QSR1¥IR[XZaUn?iZn?h?o{oNQOqOi � ?X n = 1¸�� R[QOqD�ZQSR1kG\ph?h°\pnG�]�]\poNi�L

1 = 1

� � E ~ Á : 4 7 0 ED> � 02 = Á >@>@8 4 µ Á EJ´ L?®¯QOX�hGR[t?abR[h°hFn?hs\ph<¸kG\C�.�Wk�QSR�¥IR[XZaUn?iZn?h?o�o ?qOiZQOo�QS�Zi � ?XR[QOh�o�R[��QS�.�.R��n ∈ N

M?� ?X!k�QSR��.R��nt&\p�/R[h���QOXW\pqS�.`

1 + 3 + . . .+ 2n− 1 = n2 M� � E ~ Á : 4 7 0 ED>@>@ACB 2 7 4�4 Lj® QOX!\Ck?k�QSR[X.R[h�\pn��D�/R[QSk?R[h�¼_R[QOi.R[h�k?R[X `C�?QOo�R[h°� qSR[QSl.tjn?h?o 2n+ 1

M�gI�� `CqOoNi�L

1 + 3 + . . .+ 2n− 1 + (2n+ 1) = n2 + (2n+ 1)

���/R[Xn2 + 2n+ 1 = (n+ 1)2

¸\pqS�.`��Zi.R[tjiWt?QSR[X!h?QSl.tFi.��\phGk?R[X.R��W\pqS�

1 + 3 + . . .+ 2n+ 1 = (n+ 1)2,

� \C�!o�R[h&\pn�k�QSRc¥IR[XZaUn?iZn?h?o � ?Xn+ 1

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n ∈ NM

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• ¼_R[Q x 6= 1R[QOhGR1�/R[qOQSR[�?QOo�R1m�`Ca�Y?qSR�^?R�r/\pt?q MGP1\ph?h�oNQOqOi � ?XW\pqOqSR

n ∈ N

1 + x+ x2 + . . . xn−1 =1− xn1− x .

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� �O³ � = 2 79=GE 4 8FE ~ 8 2�� E ~ Á : 4 7 0 E � �

� � E ~ Á : 4 7 0 ED>@=GE ��=GED´ L�P�QSRc�G`CXZabR[q<oNQOqOi � ?X n = 1¸?�¤R[QOq

1 =1− x1− x,

� R[h?hx 6= 1

M� � E ~ Á : 4 7 0 ED> � 02 = Á >@>@8 4 µ Á EJ´ LÄ®¯QOX�hGR[t?abR[h hFn?h \ph<¸DkG\C�.��k�QSR��G`CXZabR[q�� ?X�R[QOh o�R[��QS�]�.R��n ∈ N

�.iZQOa�aUi@¸&\pqS�.`�kG\C�.�

1 + x+ x2 + . . . xn−1 =1− xn1− x .

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1 + x+ x2 + . . . xn−1 + xn = 1−xn

1−x + xn

= 1−xn

1−x + xn(1−x)1−x

= 1−xn+xn−xn+1

1−x

= 1−xn+1

1−x��QSR[XW�Zi.R[tFi�\p�/R[X�o�R[h&\pn�k�QSRc�G`CXZabR[q � ?X

n+ 1M

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(1 + ω)n =

(

n

0

)

+

(

n

1

)

ω +

(

n

2

)

ω2 + . . .+

(

n

n

)

ωn

(1 + ω2)n =

(

n

0

)

+

(

n

1

)

ω2 +

(

n

2

)

ω4 + . . .+

(

n

n

)

ω2n

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2n + (−ω)n + (−ω2)n =

(

n

0

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· (1 + 1 + 1)

+

(

n

1

)

· (1 + ω + ω2)

+

(

n

2

)

· (1 + ω2 + ω4)

+

(

n

3

)

· (1 + ω3 + ω6)

MMM

+

(

n

n

)

· (1 + ωn + ω2n).

® QOX�� R@\Cl]tji.R[h�hG`_l.t

1 + ω + ω2 = 0

1 + ω2 + ω4 = 1 + ω2 + ω3 · ω = 1 + ω2 + ω = 0

1 + ω3 + ω6 = 1 + ω3 + (ω3)2 = 1 + 1 + 1 = 3

n?hGk°\pqOqOo�R[abR[QOh1 + ωi + ω2i =

0, 0 6= i mod 3

3, 0 = i mod 3

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3 ·((

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+

(

n

6

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+ . . .

)

= 2n + (−ε)n + (−ε2)n

= 2n + (−1)n(ωn + ω2n)

=

2n + (−1)n+1, 0 6= n mod 3

2n + (−1)n · 2, 0 = n mod 3

��qS�.`�oNQOqOi(

n

0

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n

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3.

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P = {2, 3, 5, 7, 11, 13, . . .}

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1 2 3 ×4 5 ×6 7 ×8 ×9 ×1011 ×12 13 ×14 ×15 ×16 17 ×18 19 ×20×21 ×22 23 ×24 ×25 ×26 ×27 ×28 29 ×3031 ×32 ×33 ×34 ×35 ×36 37 ×38 ×39 ×40

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2 = 2, 3 = 3, 4 = 22, 5 = 5, 6 = 2 · 3, 7 = 7, 8 = 23, 9 = 32, 10 = 2 · 5, . . .

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® QOX��?QOqSk?R[h�hjn?h¾k�QSRUr/\pt?q

N := p1 · p2 · . . . pn + 1

NQS�.i�h?QSl.tjiWi.R[QOqO�&\pXWk�n?X.l.t

p1, p2, . . . , pn¸?kG\�� R[Q<·DR[QOqOn?h?o{�Zi.R[i.�Wk?R[X!�WR��Zi

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p1, p2, . . . , pn��`CX@M�®¯QSk?R[X.�ZY?XZnGl]t � ½�gIP

6y8�E+7 4 7 0 E � ��5 � � ?Xx ∈ R

�.R[iZ�@R[h���QOX

Π(x) :=�Wh?��\pt?qJ�IXZQOa���\pt?qSR[h ≤ x.

2 3 5 7

1

2

3

4

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Π(x) 4 25 168 1229 9592 78498 664579 5761455

xln x 4 22 145 1086 8686 72382 620420 5428681

= 4 µ � ��5 � �#6y8 2 �c2 7 � µ�=GB+;Â>�= 4 µ�� *� iU� U� S � N U , � 6 UJ�Zb����¤� b�b��G� �W��`�W�W R �'� � �

Π(x) ∼ x

lnx, (x −→ ∞).

g�XZm_q �pXZn?h?oVLf(x) ∼ g(x), (x −→ ∞)

�/R�k?R[n?i.R[ilim

x−→∞f(x)/g(x) = 1.

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n, n+ 2, n+ 4, n+ 6, . . .

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3, 5; 5, 7; 11, 13; 17, 19; . . .

e � £ �Zi �#R�k?R o�R[X]\Ck?R�r/\pt?q�� ≥ 4��¼Fn?a�abRc��`Ch����¤R[QJ��XZQOa���\pt?qSR[h±

4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7, 12 = 5 + 7, . . .

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P�QSR��.R���°nG�Zi.R[X�hGR[h?hji�a�\ph � ��(%3) � �7) ���&�'$ � #"$[M��QO�?i R��¦R[QOhGRW�&`CXZabR[q$� ?X¤��XZQOa���\pt?qSR[h±1© R[iZX]\Cl]tji.R

f(n) := n2 +n+41M�� ?X

n = 0, 1, 2, 3, 4, 5, . . . , 40QS�Zif(n)

R[QOhGRc��XZQOa���\pt?q M&® \C��QS�ZiW\p�/R[X�a�QOif(41)

±gI���.l.tGR[QOhFi@¸/kG\C�.�!�_QSR[qSR��IXZQOa���\pt?qSR[hs�Gn?h?mFiZQS`ChG�Z� R[XZi.R��.`CqSl.tGR[X �jn&\Ck�X]\piZQS�]l.tGR[hº�D`Cq��_hG`CabR��ZQOhGkÄM&P1\C��QS�ZiR[QOh���hG�]\piZ�1��n?XWg�XZm_q �pXZn?h?o�k?R���°nG�Zi.R[X.����`Ch°ÇWq�\pab�!· QS�]l.tGk?R�l.m�RNM� ���IXZQOa���\pt?qSR[hsk?R[X��G`CXZa

2m − 1�¤R[X.k?R[h � � N W��&�<�<�3�W�@����XZQOa���\pt?qSR[hso�R[h&\ph?hji@M/P1\pa�QOi 2m − 1

Y?XZQOaQS�Zi@¸�QS�Zi!R��!hG`Ci��¤R[hGk�QOoG¸ kG\C�.�

m�.R[qO�/R[X�Y?XZQOa QS�Zi@M

m 2 3 5 7 11

2m − 1 3 7 31 127 2047

Y � N8S �\[¾oNq�\pn?�?i.RN¸k�QSRUr/\pt?q Fn = 22n+ 1

�.R[Q<�Zi.R[i.�WR[QOhGRc��XZQOa���\pt?q L

F0 = 3, F1 = 5, F2 = 17, F3 = 257, F4 = 65537

^a`8bj� N R[hFi.k?R�l.m_i.R�\p�/R[X@¸GkG\C�.�

F5 = 4294967297 = 641 · 6700417QS�Zi@M�® QOX��¤R[X.k?R[h°hG`Fl]t��.R[tGR[h<¸?��QSRcgIn?qSR[XWk�QSR��.R[h��&\pm_i.R[X

641o�R ��n?hGk?R[h�t&\pi@M

J J� J I�<MY � :OI�SX:=Y;YD: P ∗ ��Q ���VP † �WR ���

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4) ∀x 6= 0 ∃y | x · y = 1

QS�Zi�h?QSl]tji�R[X�� ?qOqOi@M

����� � �2��� � ������ � � � � �"�� �� � �6y8�E+7 4 7 0 E ³ ��5 g�QOh � � � �9o�R[h&\pnGR[X@¸?m�`Ca�a�n?i]\piZQO��R[X��WQOh?o�a�QOi

1�¦QS�Zi�R[QOhGR ¾R[h?o�R

R��nG�²\pa�abR[hya�QOi

���¤R[Q)Æ�Y R[X]\piZQS`ChGR[h+n?hGk · ¸�� ?X!k�QSRUk�QSRc�WR�l]tGR[h?X.R[o�R[qOh#� ?X!TWVCXZY/R[X�\pn ��R[XWk?R[XW��R[o�R[q8�� �¤o�R[qOi.R[h<M

! 8F79> } 798F;98�³ � � • ��R�k?R[X�TWVCXZY/R[X!QS�Zi�h&\pi ?XZqOQSl.ts\pnGl.t¾R[QOh��!QOh?oGM

• P�QSR�¾R[h?o�RZa�QOiWk?R[X ?�?qOQSl]tGR[h°��k?k�QOiZQS`Ch�n?hGk��n?qOiZQOY?qOQOmN\piZQS`Ch�QS�Zi!R[QOh°�!QOh?oGM

• P�QSR%¾R[h?o�RZ[i] := {a + bi | a, b ∈ Z} ⊂ C

a�QOick?R[h���R�l.tGR[hG`CY/R[X]\piZQS`ChGR[h ��`ChCtGR[Q�Nic�!QOh?o

k?R[X e �2`8W�W��]l.tGR[h�r/\pt?qSR[h<MG£�h Z[i]oNQOqOi���n?a © R[QS�ZY?QSR[q

(1 + 2i)(1 − 2i) = 5.

• � R[h&\pnG�]`�oNn?i!QS�ZiZ[√

2] := {a+ b√

2 | a, b ∈ Z} ⊂ R

R[QOh°�!QOh?oGM

• ¼_R[Q X R[QOhGR�� R[qOQSR[�?QOo�R ¾R[h?o�RN¸jn?hGk��.R[QR := {f : X −→ R} k�QSR ¾R[h?o�R�\pqOqSR[X R

� �¤R[XZiZQOo�R[h��Gn?h?mj�iZQS`ChGR[hs\pn��

XM�®¯QOXWk?R�¡Gh?QSR[X.R[h¾��k?k�QOiZQS`Ch�n?hGk�°n?qOiZQOY?qOQOmC\piZQS`Ch¾�.`CqSl.tGR[X��?n?h?mFiZQS`ChGR[h°Y?n?h?mFi �¤R[QS�.R@L

(f + g)(x) := f(x) + g(x)

(f · g)(x) := f(x) · g(x) ��/R[XZqSR[o�R[hs¼FQSRN¸&kG\C�]�

RR[QOh��!QOh?o{QS�Zi@¸?� R[QOq

RR[QOh¾�!QOh?obQS�Zi@M

RQS�Zi!�.`Coj\pX�R[QOh�TWVCXZY/R[X@¸ \p�/R[X

RQS�.i

hFn?XWR[QOh�T�VCXZY R[X@¸&�¤R[h?hXhjn?XWR[QOh�gIqSR[abR[hjiW�/R��ZQOiZ��i@M

® QOX��¤R[X.k?R[h�QOh°k�QSR��.R[X�¥I`CXZqSR��Zn?h?o�hG`_l.t��FQSR[qSR � R[QOi.R[X.RU�!QOh?o�R m�R[h?hGR[h?qSR[XZhGR[h<M

6y8�E+7 4 7 0 E ³ � � g�QOh g�qSR[abR[hFix ∈ R tGR[Q�Ni � � $�� � ¸/� R[h?h R��1QOh

RR[QOh

yoNQO�?i1a�QOi

x · y = 1M/® QOX

�.l]t?X.R[QO� R[h°\pnGl]ty = x−1 M?P�QSR�¾R[h?o�Rc\pqOqSR[XWg�QOh?tGR[QOi.R[h°��`Ch

RhG`CiZQSR[X.R[h���QOX

E(R) := {x ∈ R | ∃y ∈ R : x · y = 1}.

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x1 · x2R[QOhGRcg�QOh?tGR[QOi@M

! 8F79> } 798F;98�³ � ³ • ¼FQOhGk�\pqOqSR x 6= 0g�QOh?tGR[QOi.R[h<¸�.`�QS�Zi

RR[QOh�TWVCXZY/R[X@MG��qS�.`

RQS�Zi!R[QOh�T�VCXZY R[X ⇔ E(R) = R \ {0}

• E(Z) = {1,−1}

• E(Z[i]) = {1, i,−1,−i}

• P R[XW�WQOh?o Z[√

2]t&\pi ��#$���� �"��� $ � $!g�QOh?tGR[QOi.R[h<M

(√

2− 1)(√

2 + 1) = 1,

\pqS�]`�QS�Zi √2− 1

R[QOhGR1g�QOh?tGR[QOi@M/¼_`Ca�QOiW�ZQOhGk�\pnGl]t�k�QSRc�+`Ci.R[h?�@R[h���`Ch √2− 1

g�QOh?tGR[QOi.R[h�L

(√

2− 1)2 = 3− 2√

2, (√

2− 1)3 = 5√

2− 7, . . .

6y8�E+7 4 7 0 E ³ ��¶ ¼_R[QRR[QOh �!QOh?o�n?hGk

a, b ∈ R MÄ® QOXU�]\po�R[h a � $ � ���b¸ ��`Ci]\piZQS`Ch

a | b ¸Ä� R[h?h R���R[QOhq ∈ R oNQO�?i!a�QOi

a · q = b

·<XZQ/iWk�QSR���h?QSl]tji���n<¸G�.`��.l]t?X.R[QO�?iWa�\pha - b

M

= 4 µ�³ �SK � � 8_79;9H+= 2 :8F7 4 > 2 8j´8F;�E � ��"0/�(*>?+

� @ 1 | a ;5 < a | a @�� &�(?+L& : /�(*>?+ a | 0 @� @ � " a | b ;5 < b | c 7 � > /�+ a | c @� @ � " a | b ;5 < c | d 7 � > /�+ ac | bd @� @ � " a | b ;5 < a | c 7 � > /�+ a | (b+ c) @� @ � " a | b ;5 < r ∈ R 7 � > /�+ a | rb @

DE&GF�&�(H" @ g�QOh�� \Cl.tGRa ��?n?h?o��]\pn���oj\p� RNMj© R[abR[XZm�RN¸FkG\C�.�Ik�QSR�qSR[iZ��i.R��WR[o�R[qGR[QOh�¼FY/R[��Q�\pq� \pqOqk?R[X¦k�XZQOiZi.R[h���R[o�R[qQS�Zi@M ½�gIP

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����� � � �2� ����� � � � ��� � �������� �� �2 � �;� ��� � � �?� ����� ��� � � �

·JR[QOqOn?h?o{QS�Zi�QOhZ��� \pX�h?QSl.tFiWQOa�abR[X!abVCoNqOQSl]t<¸\p�/R[X!R���oNQO�?i �#$�� �-�� � % ��� � $$)��TL

7

2= 3 +

1

2, 7 = 3 · 2 + 1

= 4 µ�³ �9à � � 8_79; Á EJ´ � 7 4�� 8j> 4 � ��&�( &�5a, b

5 + 8;: >*(�� $'&��� $N> &�5 @ �! 5N5 /�(213+%&." &�(*5 < &�N+ ( / 1,&." + (���

+L&

q, r ∈ N0 := N ∪ {0} � (?+

a = q · b+ r

�(?+

0 ≤ r < b @DE&GF�&�(H" @ ®¯QOX �/R[�¤R[QS�.R[h k�QSR�g�^_QS�Zi.R[h?�b�]`CqSl.t�R[QOhGR[XUP1\pX.�Zi.R[qOqOn?h?oya�QOiU£ hGk�n?m_iZQS`Chºh&\Cl.t a MÄ£ �Zi a < b

¸Ä�]`�.R[iZ�@R

q = 0, r = a¸jn?hGkb��QOX �ZQOhGk � R[XZiZQOoGMj£ �Zi

a ≥ b ¸j�.`1QS�Zi a′ := a− b mFqSR[QOhGR[X \pqS� a n?hGk�a�QOi¤£ hGk�n?mFiZQS`Chm�VCh?hGR[h���QOX!��`CX]\pnG�]�.R[iZ�@R[h<¸&kG\C�.��k�QSRU© R[t&\pn?Y?iZn?h?o�� ?XaoNQOqOi@MG��qS�.`

a′ = q′b+ r

a�QOi0 ≤ r < b

M?P \ph?h�QS�ZiW\p� R[X

a = a′ + b = (q‘b+ r) + b = (q′ + 1)b+ r

��qS�.`Ut&\p�/R[hy��QOX a�QOiq = q′ + 1

k�QSR P \pX.�.i.R[qOqOn?h?o � ?XaM�Ç�aÈk�QSR�gIQOhGk?R[n?iZQOoNm�R[QOi!��n��@R[QOo�R[h<¸�� R[iZX]\Cl]tji.R[h

��QOX!��� R[QJP1\pX.�Zi.R[qOqOn?h?o�R[h

a = qb+ r,

a = q′b+ r′.

P�n?X.l.t����?��QSR[tGR[h°R[XZt&\pqOi.R[h���QOX(q′ − q)b = r − r′.

P1\r − r′ < b

¸�� `CqOoNir = r′

n?hGkq = q′

M ½�gIP® QOX!�.l.t?X.R[QO�/R[h

r = a mod b

�9o�R��ZY?X.`_l.tGR[ha% !+�(����!

b��M?�Wn ��R[X.k?R[a m�VCh?hGR[h���QOXW�.l]t?X.R[QO� R[h�L

q = babc.

� ?XWR[QOhGR X.R�R[qOqSR�r/\pt?qxQS�ZiWkG\p�/R[Q bxc k�QSR1oNX.V"�Ni.R1oj\ph?�@R�r/\pt?q<mFqSR[QOhGR[X�`Fk?R[X�oNqSR[QSl]t x M

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6y8�E+7 4 7 0 E ³ � � ¼_R[Qa ∈ N

MT (a) := {c ∈ N : c | a}

QS�Zi!k�QSR�¾R[h?o�Rc\pqOqSR��9Y/`N�ZQOiZQO��R�� ·DR[QOqSR[XW��`ChaM ¼_R[QSR[h

a, b ∈ NM

T (a, b) := T (a) ∩ T (b)

QS�Zi!k�QSR�¾R[h?o�R1k?R[X!o�R[abR[QOhG�]\pabR[h¾·JR[QOqSR[X���`Chan?hGk

bM¼_R[iZ�@R

ggT (a, b) = max{c : c ∈ T (a, b)}.

ggT(a, b)tGR[Q�NiWk?R[X �"�� �� � $ �'$ % $ � ) �(% $ � $�� �($�����`Ch

an?hGk

bM

= 4 µ�³ � � ��&�( &�5a, b ∈ N �

(?+a ≥ b @ � ( : 798 $ : &�5���&�(*> ;53/ � (?+�� &."G+ < : � $ @

a = q · b+ r,

F�1,&�( < & : �!&." + r = (a mod b) ∈ {0, 1, 2, . . . , b− 1} (H" + @ �! 5N5 /�(*>?+

T (a, b) = T (r, b).

� 5,"G1N&."�5 < & : &%/�(*>?+

ggT(a, b) = ggT(r, b).

DE&GF�&�(H" @ £ �Zi c ∈ T (a, b)¸Ä�.`{oNQOqOi

c | a n?hGk c | b M P1\ph?h¾QS�.i1\p�/R[Xc\pnGl.t c ·JR[QOqSR[X1��`Ch a − q · b = r¸�]`

kG\C�.�c ∈ T (r, b)

M £ �Zi�n?a�o�R[m�R[t?XZic ∈ T (r, b)

¸ �.`�oNQOqOic | b n?hGk c | r ¸?n?hGk��.`Ca�QOiWQS�Zi�\pnGl.t c R[QOh¾·JR[QOqSR[X��`Ch

q · b+ r = a¸?kÄM t<M

c ∈ T (a, b)M ½�gIP

= 4 µ�³ ��5�� � ��Á :<;�7 ~ 79>@ACBJ8 2{� ;9´ 0&2 7 4 B �sÁ > � ��(*53/ 31,& - a, b ∈ N �(?+a ≥ b @

• a1 := a, a2 := b

• � 8;: i = 2, 3, . . ." &G+��;&

ai+1 := ai−1 mod ai @

• ��+ � � ��6 F�&�5N5 ak+1 = 0 @�! 5N5 /�(*>?+

ak = ggT(a, b).

DE&GF�&�(H" @

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• P R[X1�WqOo�`CXZQOiZt?a�nG� i.R[XZa�QOh?QSR[XZi�L/®¯QOX R[XZt&\pqOi.R[hºR[QOhGR��ZiZX.R[h?o{ab`ChG`Ci.`Ch � \pqOqSR[hGk?R��G`CqOo�R�h?QSl.tFi1hGR[oj\��iZQO��R[XWoj\ph?�@R[X�r/\pt?qSR[h

a1 > a2 > a3 > . . .

� \Cl.t�tGVFl]tG�Zi.R[hG�a1¼_l.t?XZQOiZi.R[h°QS�Zi!k�QSR

0R[XZX.R[QSl.tFi@M ��£ hy®¯QOXZm_qOQSl.t?m�R[QOi�o�R[tjiWR��!�FQSR[q<�.l]t?hGR[qOqSR[X@M �

• P R[XW��qOo�`CXZQOiZt?aUnG� �9n?h?m_iZQS`Ch?QSR[XZi!m�`CXZX.R[m_i�LGgI��QS�.i

T (a1, a2) = T (a2, a3) = . . . = T (ak, 0) = T (ak).

¼_`Ca�QOiWQS�Zi

ggT (a1, a2) = ggT (a2, a3) = . . . = ggT (ak, 0) = ak.

® QOXJ�.R[tGR[hc�.`Coj\pX<abR[t?X@¸�h��pa�qOQSl.t�L � ;�;98c´&8 � 8_79ED>@= � 8FE � 8F7�;98 2 ��`Chan?hGk

b�ZQOhGk1·DR[QOqSR[XJ��`Ch

ggT (a, b)M

½�gIPs\phUmN\ph?h�k?R[h�gIn?mFqOQSk�QS�.l]tGR[h���qOo�`CXZQOiZt?aUnG��\pnGl]t�\pqS�+R[QOhGR � $ ��� $ � !"��#$�� �-�� �'$� % ��� � $$)��_\pn����]l.t?X.R[QO�/R[h�L

a1 = q1 · a2 + a3

a2 = q2 · a3 + a4MMMak−2 = qk−2 · ak−1 + ak

ak−1 = qk−1 · ak + 0

! 8F79> } 798F;98�³ �955

LNM�® QOX!�6�pt?qSR[ha = a1 = 42

¸b = a2 = 30

M&P1\ph?h

42 =1 · 30 + 12

30 =2 · 12 + 6

12 =2 · 6 + 0.

® QOXWR[XZt&\pqOi.R[hggT(42, 30) = 6

M

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¶�� ³ 6 ��� �����¾À 6 ���¹Ày�{�¾��� � �I�{���¦���

e�M�® QOX!�6�pt?qSR[ha = 11220

n?hGkb = 5187

M

11220 = 2 · 5187 + 846

5187 = 6 · 846 + 111

846 = 7 · 111 + 69

111 = 1 · 69 + 42

69 = 1 · 42 + 27

42 = 1 · 27 + 15

27 = 1 · 15 + 12

15 = 1 · 12 + 3

12 = 4 · 3 + 0.

��qS�.`�QS�ZiggT(11220, 5187) = 3

M = 4 µ�³ ��5 � �#6y= 2 > 4 8F;�; Á ED´&>@>�= 4 µ � �D2 ~ 8FE ´ 2p|��4 8FE¯´&8 � 8F7�ED>@= � 8_E � 8F7�;98 2 ���&�( &�5

a, b ∈ N �(?+a > b @ �! 5N5 /�(213+0&."

A,B ∈ Z 6 " ��< #"�"

ggT (a, b) = A · a+B · b.� 5 5'&�5N5 +

A;5 < B < ( & ©��[�@`Cn?i �!� �FTW`_R�ªb��QSR[hji.R[h � 5 a ;5 < b @

¥I`CXWk?R[a © R[� R[QS�Wk?R���¼�\piZ�@R�� ?� R[XZ�@R[n?o�R[h���QOXWn?hG�W\ph?t&\phGky��`Ch�© R[QS�ZY?QSR[qSR[h���`Ch°�.R[QOhGR[X�� ?qOiZQOoNm�R[QOi@M

! 8F79> } 798F;98�³ �95 � ® QOX��]l.t&\pnGR[h¾n?hG��k�QSR��.l.tG`Chsk�n?X.l.t?o�R � ?t?XZi.R[h�© R[X.R�l.t?hFn?h?o�R[hggT(42, 30) = 6

n?hGkggT(11220, 5187) = 3

hG`_l.t?a�\pqS��\ph<MLNM

42 =1 · 30 + 12

30 =2 · 12 + 6

12 =2 · 6 + 0.

P�n?X.l]t°� Gl]mF� \pXZi.�#��g�QOhG�.R[iZ�@R[h°� R[m�`Ca�abR[h°��QOX

6 =1 · 30− 2 · 12=1 · 30− 2(42− 1 · 30)=− 2 · 42 + 3 · 30.

��qS�.`�QS�Zi�QOh°k�QSR��.R[a �&\pqOqA = −2

n?hGkB = 3

MJ � ( @�<M:=Y;YD: . �!*9P &D@BP ∗ � � �Y+P † � � �

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³ � � � 8F7�; Á EJ´ � 7 4b� 8F> 4 � ��Á :<;�7 ~ 79>@ACBJ8 2�� ;9´ 0&2 7 4 B �sÁ > ¶$�

e�M

11220 = 2 · 5187 + 846

5187 = 6 · 846 + 111

846 = 7 · 111 + 69

111 = 1 · 69 + 42

69 = 1 · 42 + 27

42 = 1 · 27 + 15

27 = 1 · 15 + 12

15 = 1 · 12 + 3

12 = 4 · 3 + 0.® QOX!X.R�l]t?hGR[h���QSR�k?R[X!X Gl]mF�6�pXZi.��L

3 = 15− 1 · 12= 15− 1 · (27 − 1 · 15) = −1 · 27 + 2 · 15= −1 · 27 + 2(42 − 1 · 27) = 2 · 42 − 3 · 27= 2 · 42− 3(69 − 1 · 42) = −3 · 69 + 5 · 42= −3 · 69 + 5(111 − 1 · 69) = 5 · 111− 8 · 69= 5 · 111 − 8(846 − 7 · 111) = −8 · 846 + 61 · 111= −8 · 846 + 61(5187 − 6 · 846) = 61 · 5187 − 374 · 846= 61 · 5187 − 374(11220 − 2 · 5187) = −374 · 11220 + 809 · 5187.

��qS�.`A = −374

n?hGkB = 809

MDE&GF�&�(H" @ ®¯QOX��@R[QOo�R[h°a�QOi!£�hGk�n?m_iZQS`Ch<¸GkG\C�.�W\pqOqSR ai R[QOhGRcP1\pX.�Zi.R[qOqOn?h?o{k?R[X!�&`CXZa

ai = Ai · a+Bi · b�/R��ZQOiZ�@R[h<MN® QOXI�.R[iZ�@R[h

A1 = 1, B1 = 0n?hGk

A2 = 0, B2 = 1¸��.`1kG\C�.�Ik�QSR���nG�.�]\po�R � ?X

i = 1, 2�ZiZQOa�a�i@M

��h?o�R[hG`Ca�abR[h<¸?��QOX t&\p� R[h�hjn?h��]`�R[QOhGR1P1\pX.�Zi.R[qOqOn?h?o � ?Xa1, a2, . . . , ai+1

o�R ��n?hGk?R[h<Mai+2

R[hji.�Zi.R[tFi�\pqS��WR��.i

ai = qi · ai+1 + ai+2.

¼_R[iZ�@R[h���QOXWk�QSRUP \pX.�.i.R[qOqOn?h?ob��`Chain?hGk

ai+1R[QOh<¸G�.`�¡GhGk?R[h���QOX

ai+2 =(Ai · a+Bib)− qi(Ai+1a+Bi+1b)

(Ai − qi ·Ai+1)a+ (Bi+1 − qi ·Bi+1)b

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K$� ³ 6 ��� �����¾À 6 ���¹Ày�{�¾��� � �I�{���¦���

® QOX�m�VCh?hGR[h¾\pqS�.`Ai+2 = Ai − qi · Ai+1, Bi+2 = Bi+1 − qi ·Bi

�.R[iZ�@R[h<¸?n?hGk�a�QOi!k?R[a ��XZQOh?��QOY°k?R[XW��`CqOqS�Zi�phGk�QOo�R[h�£ hGk�n?mFiZQS`Ch�QS�.iW\pqOqSR��!�/R[��QSR��.R[h<M ½�gIPP R[X�© R[� R[QS��oNQO�?i�R[QOhGR[h �WqOo�`CXZQOiZt?a�nG�@¸�k?R[h 8 2�3 8F7 4 8 2�4 8_E � Á :<;97 ~ 79>@ApBD8FE � ;9´ 02 7 4 B �sÁ > ¸+k?R[X�R[QOhGR ?�/R[X.�ZQSl.tFiZqOQSl.tGR�© R[X.R�l]t?hjn?h?o�k?R���oNo�· n?hGk�k?R[X!©��[�@`Cn?i#��T�`FR�ªb��QSR[hFi.R[h

An?hGk

BR[XZabVCoNqOQSl.tFi@M

q 11220 5187

11220 1 0

5187 2 0 1

846 6 1 −2

111 7 −6 13

69 1 43 −93

42 1 −49 106

27 1 92 −199

15 1 −141 305

12 1 233 −504

3 4 −374 809

P�QSR°��nG�Zoj\ph?o��Z��\pt?qSR[h � R[X.k?R[h qOQOh?m���n?hji.R[X.R[QOh&\phGk?R[Xyo�R��.l]t?XZQSR[� R[h<¸�kG\pXZn?hFi.R[XykG\ph?h k�QSRs\pn���iZX.R[i.R[hGk?R[h�WR��.i.RNM�£�h{k?R[X¦���¤R[QOi.R[h�¼FY&\pqOi.R!� R[X.k?R[h�k�QSR�R[hG�ZiZY?X.R�l.tGR[hGk?R[h�½�nG`CiZQSR[hFi.R[h

q\pn���o�R[iZX]\po�R[h<Mj£ h{k?R[X¦k�XZQOiZi.R[h

n?hGk �_QSR[XZi.R[h ¼FY&\pqOi.R�¡GhGk?R[h¯��QOX�k�QSR�TW`_R�ªb��QSR[hji.R[h<¸¤a�QOi�k?R[hGR[h a�\ph¯k�QSR°r/\pt?qSR[h¯k?R[X�R[X.�.i.R[h¯¼FY&\pqOi.R\pqS�b�ÄQOhGR@\pXZm�`Ca��?QOh&\piZQS`Ch k?R[Xb�WnG�Zoj\phG�.��\pt?qSR[h¯\pnG�.k�X Gl.m�R[h¯mC\ph?h<M¦P�QSR��]R�¼FY&\pqOi.R[h mC\ph?h a�\ph � R��FnGR[a�¤R[QOi.R[X�� ?t?X.R[h<¸/QOhGk?R[a a�\ph¾k�QSR � !"���$ � � � $ % �"�'� − q · �($ � � � $ % �"�'�&R[QOhjiZX�poNi@MÄ£ h°k?R[X�qSR[iZ��i.R[h�r R[QOqSRc¡GhGk?R[h��QOXWkG\ph?h�k?R[hyoNo�·un?hGk�k�QSR1��n?o�R[tGVCXZQOo�R[h°©��[�@`Cn?i#��T�`FR�ªb��QSR[hFi.R[h<M

= 4 µ�³ ��5F³ � �c2 7 � 8F79´&8_EJ>�ApBD= � 4 � 0 E �c2 7 � µ�=GB+;98FE � ��&�(p&�(*5'& � : ( � � $N> ;5 < p | a · b @ � 5N5 /�(*>?+

p | a � < & : p | b @DE&GF�&�(H" @ ¼_R[Q c := ggT (a, p)

MDP1\pR[QOhGRb��XZQOa���\pt?qIQS�Zi@¸/oNQO�?i�R���� ?X

c��� R[Q4¾VCoNqOQSl.t?m�R[QOi.R[h�LJgIhji�� R�k?R[X

c = p`_k?R[X

c = 1M&£ �Zi

c = p¸?�.`�oNQOqOi

p | a MG£ �Zi c = 1¸�.`�oNQO�?i�R��!h&\Cl.t°k?R[a P \pX.�.i.R[qOqOn?h?o��.�]\piZ� � ?XWk?R[h

ggT An?hGk

B¸?�.`�kG\C�.�

1 = A · a+B · p.P�n?X.l.t�°n?qOiZQOY?qOQOmN\piZQS`Ch�a�QOi

bR[XZt&\pqOi.R[h���QOX!t?QSR[X]\pnG�

b = A · ab+B · pb.

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³ � � 6y8 2��<Á E ~ = � 8FE 4 =G;9>@= 4 µ ~ 8 2 8F;98 � 8FE 4 = 2 8FE � =GBD;98FE 4 BD8 02 798 K<5

P1\p | ab � `CqOoNi!t?QSR[X]\pnG� p | b ¸��¤`Ca�QOi�k�QSRc��nG�.�]\po�R �/R[��QSR��.R[hyQS�Zi@M ½�gIP

��QSR[X]\pnG��qSR[QOi.R[i!a�\ph�a�QOiW£ hGk�n?m_iZQS`Ch�R[QOh�� \Cl]t°\p��L

= 4 µ�³ ��5�¶

p | a1 · a2 · . . . · ak ⇒ p | ai 7L8;: &�(*5 i ∈ {1, 2, . . . , k}.

g�QOhGR1��QSl]tjiZQOo�R��WhF�¤R[hGk�n?h?o{QS�Zi�L

= 4 µ�³ ��5_K � "G+p ��: ( � ;5 < k ∈ {1, 2, . . . , p− 1} 6 " � /�(*>?+

p |(

p

k

)

DE&GF�&�(H" @ �WnG�!k?R[X�P�R�¡Gh?QOiZQS`Ch°k?R[XW© QOhG`Ca�Q�\pqOm�`FR�ªb��QSR[hFi.R[h � `CqOoNi�L

p · (p− 1) · . . . · (p− k + 1) = k! ·(

p

k

)

P�QSR�qOQOh?m�RU¼_R[QOi.R QS�Zi i.R[QOqO�&\pX!k�n?X.l.tp¸?\p� R[X�kG\

k < pQS�Zi@¸_QS�Zi

k!h?QSl.tFi!i.R[QOqO�&\pX!k�n?X.l.t

pM?¼_`Ca�QOi�aUnG�]� (p

k

)

k�n?X.l.tpi.R[QOqO�&\pX��.R[QOh<M ½�gIP

����� � �������������������� ���2�;��� � �� � ����� ������� ������� � ����������������� �*���£ a �J\phGk � $� ���boNQO�?i�R��chFn?XUo�R[X]\Ck?Ryr/\pt?qSR[h<¸<n?h?o�R[X]\Ck?Ryr/\pt?qSR[h �ZQOhGk R[QOh�� \Cl]t ��R[XZ�/`Ci.R[h<MDg�QOhGR�r/\pt?qtGR[Q�Ni C ��� %�� ��\pt?q ¸�� R[h?h°�ZQSR h?QSl]tji�\pqS�W��X.`Fk�n?m_i!m_qSR[QOhGR[X.R[X!o�R[X]\Ck?R[X�r/\pt?qSR[h¾kG\pX.�Zi.R[qOqO�&\pX!QS�Zi@M¼_`Ca�QOi��ZQOhGk

2, 6, 10, 14, 18, . . .

n?hGk°\pqOqOo�R[abR[QOhGR[Xc\pqOqSR�r/\pt?qSR[h¾k?R[X��G`CXZa2(2k + 1)

��XZQOa�� ��\pt?qSR[h<Ml��R�k?R�r/\pt?qJQS�ZiW�IX.`Fk�n?m_i���`Chº��XZQOa����\pt?qSR[h<¸&\p�/R[XWk�QSRUr R[XZqSR[oNn?h?o�QS�.i�h?QSl.tji�R[QOhGk?R[n?iZQOoVL

60 = 6 · 10 = 2 · 30, 180 = 18 · 10 = 6 · 30, R[i.lNM

·JR�l]t?h?QS�.l.t�o�R��ZY?X.`Fl]tGR[h��?QOqSk?R[h�k�QSR¤o�R[X]\Ck?R[h�r/\pt?qSR[h�R[QOhGR[h��WQOh?o�`Ct?hGR1��g�qSR[abR[hji@Mj�W�/R[X)\pnGl]t�QOh��WQOh?o�R[h

a�QOi1QS�Zi�r R[XZqSR[oNn?h?o{QOh�QOXZX.R�k�n?��QO�?qSRUg�qSR[abR[hFi.RU` ��i�h?QSl.tFi�R[QOhGk?R[n?iZQOoGM

6y8�E+7 4 7 0 E ³ ��5&à ¼_R[QRR[QOhs�!QOh?o�n?hGk

r ∈ R R[QOhsg�qSR[abR[hji@MrtGR[Q�Ni � � � $B�(� � ��! $�� ¸ �¤R[h?hº\pnG�

r = a · b� `CqOoNi@¸GkG\C�.�

a`Fk?R[X

bR[QOhGRcg�QOh?tGR[QOi�QS�Zi@M

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K � ³ 6 ��� �����¾À 6 ���¹Ày�{�¾��� � �I�{���¦���

P�QSR QOXZX.R�k�n?��QO�?qSR[h¾gIqSR[abR[hji.R�\pnG�Z�ZQOhGk ±p a�QOi p R[QOhGRU��XZQOa���\pt?q M£ h�k?R[a �!QOh?o

Z[√−5]

oNQOqOiW��n?a © R[QS�ZY?QSR[q

2 · 3 = (1 +√−5) · (1−

√−5),

\p�/R[X!k�QSRcg�qSR[abR[hji.R2, 3, (1 +

√−5), (1 −

√−5)

�ZQOhGk�\pqOqSR QOXZX.R�k�n?��QO�/R[q M® \pXZn?a QS�Zi!k?R[h?h°R[QOo�R[hjiZqOQSl.t

13 · 29 6= 11 · 31 ?

P R[X ��XZn?hGk{QS�Zi�LFk�QSR!qOQOh?m�R�¼_R[QOi.R�QS�Zi¤i.R[QOqO�&\pX k�n?X.l]t13¸_�.`Ca�QOi¦a G�.�Zi.R

11 ·31 i.R[QOqO�&\pX k�n?X.l.t 13�.R[QOh<¸_\pqS�.`

a G�]�Zi.RcR[hji �¤R�k?R[X11`Fk?R[X

31k�n?X.l]t

13i.R[QOqO�&\pXW�]R[QOh<¸?� \C�Wh?QSl.tji�k?R[X!�&\pqOq<QS�Zi@M&P�QSR�� � ?t?XZi���n?a

= 4 µ�³ ��5�� � �<Á E ~ = � 8_E 4 =G;Â>�= 4 µ ~ 8 2�� =GB+;Â8_E 4 BJ8 0&2 798 � & < & �� $N> n ∈ N(H"G+� 7 R[QOhGk?R[n?iZQOo�R1®�R[QS�.R

� " �L$ : &�(21N&�5 >H" � : � < ;4�+ � 5 � : ( � � $N> &�5 @DE&GF�&�(H" @ �QOi!£ hGk�n?mFiZQS`Ch�h&\Cl.t n M?� ?X n = 1

QS�Zi!k�QSR1��nG�.�]\po�R XZQSl.tFiZQOoGM&��h?o�R[hG`Ca�abR[h<¸&k�QSRc��nG�.�]\po�R1�]R[QXZQSl.tFiZQOo � ?XW\pqOqSR�r/\pt?qSR[h

< nM�® QOX!�¤`CqOqSR[h��@R[QOo�R[h<¸&kG\C�.�W\pnG�

n = p1 · p2 · . . . · pk = q1 · q2 · . . . · qla�QOi

k ≥ 2¸l ≥ 2

n?hGkp1 ≤ p2 ≤ . . . ≤ pk

��XZQOa���\pt?qSR[h,

q1 ≤ q2 ≤ . . . ≤ ql�IXZQOa���\pt?qSR[h

� `CqOoNi@¸?kG\C�.�k = l

n?hGkpi = qi

M�P1\p1k�QSR qOQOh?m�RU¼_R[QOi.R i.R[QOqOi@¸�oNQOqOi

p1 | q1 · q2 . . . qln?hGky�]`Ca�QOi

p1 | qi� ?X

R[QOhiMP1\

p1n?hGk

qi��XZQOa���\pt?qSR[hº�ZQOhGkĸ$��`CqOoNi

p1 = qiMÄ� R[h&\pns�.`�R[XZt&\pqOi.R[hs��QOX

q1 = pj� ?X�R[QOh

jM P1\

\p�/R[Xp1 ≤ pj = q1 ≤ qi = p1,

� `CqOoNip1 = pj = qi = q1

M?®¯QOX�m�VCh?hGR[h�hjn?hp1 = q1

\pnG�!k?R[X���qSR[QSl]tjn?h?o�� R[oNi.R[QOqSR[h°n?hGkyR[XZt&\pqOi.R[h

n/p1 = p2 · p3 · . . . · pk = q2 · q3 · . . . · ql.

P1\n/p1 < n

¸�� `CqOoNiW\pnG�Wk?R[X!£ hGk�n?mFiZQS`ChG�Z��`CX]\pnG�]�.R[iZ��n?h?oG¸kG\C�]�k = l

n?hGkpi = qi

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6y8�E+7 4 7 0 E ³ ��5�� P1\C� # �($�� )�� $ �'$ % $�� ) �(% $ � ��$ � � � ��� $U��`Cha ∈ N

n?hGkb ∈ N

QS�Zi

kgV (a, b) := min{c ∈ N : a | c n?hGkb | c}

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³ �O³Z[i]

Á E ~ ÄÁ+� � 8FE � 0 E "�Á = ~ 2 = 4 8_E K �

gI��oNQOqOiggT (a, b) · kgV (a, b) = a · b.

£ �Zia = pα1

1 · pα2

2 · . . . · pαk

k

n?hGkb = pβ1

1 · pβ2

2 · . . . · pβk

k ,

�.`�QS�ZiggT (a, b) = pγ11 · p

γ22 · . . . · p

γk

k ,

kgV (a, b) = pδ11 · pδ22 · . . . · pδkka�QOi

γi := min(αi, βi), δi := max(αi, βi).

! 8F79> } 798F; ³ � ���

90 = 2 · 32 · 5,84 = 22 · 3 · 7,

ggT (90, 84) = 2 · 3 = 6,

kgV (90, 84) = 22 · 32 · 5 · 7 = 1260.

�����Z[i]

�������� � ����� ������� ���������� ���®�R[qSl.tGRyr/\pt?qSR[h q�\C�]�.R[h �ZQSl]t \pqS��¼Fn?a�abR{��`Ch����¤R[Q!½�n&\Ck�X]\pi.R[h kG\pX.�.i.R[qOqSR[h±{®¯QOXcY?X.`C�?QSR[X.R[h ��n?h��Cl.tG�.iR[QOh?a�\pqJR[QOh?QOo�R�r/\pt?qSR[hs\pnG��n?hGkyqSR[o�R[h�R[QOhGRU·D\p�/R[qOqSRU\ph�L

1 2 3 4 5 6 7 8

12 + 02 12 + 12 22 + 02 22 + 12 22 + 22

9 10 11 12 13 14 15 16

32 + 02 32 + 12 32 + 22 42 + 02

17 18 19 20 21 22 23 24

42 + 12 32 + 32 42 + 22

= 4 µ�³ � �/5 ��(*5'&E5 + 8;: >*(��L$'& �� $N> < & : � � :�� 4k + 3(H" + 43&�(*5'&��

���& �5 ��F�&�(�� <�: +L&�5 @

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KG³ ³ 6 ��� �����¾À 6 ���¹Ày�{�¾��� � �I�{���¦���

DE&GF�&�(H" @ �Wh?o�R[hG`Ca�abR[h<¸ 4k + 3 = a2 + b2M<P1\�k�QSR�qOQOh?m�R{¼_R[QOi.R�n?h?o�R[X]\Ck?R�QS�Zi@¸/oNQOqOiUkG\C�U\pnGl]t � ?Xck�QSR

X.R�l.tFi.R{¼_R[QOi.RNM<��qS�.`�aUnG�.�ao�R[X]\Ck?R�n?hGk

bn?h?o�R[X]\Ck?Rb�.R[QOh�`_k?R[X�n?a�o�R[m�R[t?XZi@MJ®¯QOX m�VCh?hGR[h \ph?hGR[t?abR[h<¸

kG\C�.�ao�R[X]\Ck?R1QS�Zi@¸&\pqS�]`

a = 2cn?hGk

b = 2d+ 1MG®¯QOX��/R[m�`Ca�abR[h

4k + 3 = (2c)2 + (2d+ 1)2¸&\pqS�.`

4(k − c2 − d2 − d) = −2.

P�QSR qOQOh?m�R�¼_R[QOi.R1QS�Zi�k�n?X.l.t���i.R[QOqO�&\pX@¸&h?QSl.tji \p� R[X!k�QSRcX.R�l.tji.R�¼_R[QOi.R@�GR[QOh�® QSk?R[X.�ZY?XZnGl.t<M ½�gIP

= 4 µ�³ � � � ��&�( &�5a 6 b 6 c 6 d / 5 �;& �� $N> &�5 @ �! 5N5 /�(*>?+

(a2 + b2)(c2 + d2) = (ac− bd)2 + (ad+ bc)2.

DE&GF�&�(H" @ ¼_R[iZ�@R z = a+ ib¸w = c+ id

M&© R[abR[XZm�R@L |z|2 = (a2 + b2)¸ |w|2 = (c2 + d2)

n?hGky�.`Ca�QOi

(a2 + b2)(c2 + d2) = |z|2|w|2 = |z · w|2 = (ac− bd)2 + (ad+ bc)2.

½�gIP

z{0&2p0 ;�;9= 2 ³ � � � � (*5 < n ;5 < m � &GF�&�(*>H"�� ��� & ��F�&�( & : � <�: +L& 6 " � (H" +� � $ n ·m � ���

& ��F�&�( & :� <�: +L& @! 8F79> } 798F; ³ � �?³

(22 + 32) · (32 + 12) = 32 + 112

= 4 µ�³ � �&¶ � � 8 2C� = 4 � ��(*5'&�� : ( � � $N> < & : � � :�� p = 4 · k + 1(H" + < : " +L&�>*>21� : >H" � ��� & � 5 ��F�&�(

� <�: +L&�5 @® QOX�� R[X.k?R[h k�QSR��.R[h ¼�\piZ�yQOh abR[t?X.R[X.R[h¯¼_l.t?XZQOiZi.R[h �/R[�¤R[QS�.R[h<MIrn?h��Cl]tG�Zib�/R[hGVCiZQOo�R[h ��QOXb·DR[QOqOn?h?oºa�QOi�WR��.i�QOh

Z[i]� \ph&\pqS`Cob��n?XW·DR[QOqOn?h?o�a�QOi!�WR��.i�QOh

Z��M

• ® QOXWk?R�¡Gh?QSR[X.R[h δ : Q[i] −→ Q; z 7→ δ(z) := |z|2 M?��qS�.` δ(a + ib) = a2 + b2 > 0MG£ hG�Z� R��]`ChGk?R[X.R

QS�.iδ(z) ∈ N0

� ?Xz ∈ Z[i]

M

• gI�WoNQOqOi δ(z · w) = δ(z) · δ(w)¸&k�QSRc�?n?h?m_iZQS`Ch

δQS�Zi %'��� ���65 � �7#&� ��� � M

• ¼FQOhGk z = a+ ib¸w = c+ id

QOhZ[i]

a�QOiw 6= 0

¸G�.`�QS�.i

z

w=a+ ib

c+ id=: α+ iβ ∈ Q[i].

• ® �pt?qSR1hjn?hu, v ∈ Z

a�QOi |α− u| ≤ 12

¸ |β − v| ≤ 12

M

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³ �O³Z[i]

Á E ~ ÄÁ+� � 8FE � 0 E "�Á = ~ 2 = 4 8_E K¶

• ¼_R[iZ�@R q := u+ ivMGg¦��oNQOqOiWkG\ph?h

z

w= q + ((α − u) + (β − v)i),

�]`�kG\C�.�z = q · w + r

a�QOir = ((α− u) + (β − v)i)w.

• gI� ��`CqOoNiδ(r) = ((α− u)2 + (β − v)2)δ(w) ≤ (

1

4+

1

4)δ(w) ≤ 1

2δ(w)

� Á :<;�7 ~ 79>@ACBD8 2°� ;9´ 02 7 4 B �¾Á >y7�E Z[i] � ¼_R[QSR[h z, w ∈ Z[i]n?hGk

w 6= 0MÄP�n?X.l.t���QSR�k?R[XZtG`CqOi.R{·JR[QOqOn?h?o

a�QOiW��R��Ziz1 = q1 · z2 + z3

z2 = q2 · z3 + z4MMMzk−2 = qk−2 · zk−1 + zk

zk−1 = qk−1 · zk + 0R[XZt&\pqOi.R[h���QOX!R[QOhGRU�G`CqOo�R

z1 = z, z2 = w, z3, . . .a�QOi�k?R[XWg�QOo�R[hG�.l]t&\ �9i@¸kG\C�.�

δ(z2) > δ(z3) > δ(z4) > . . . ,

�.`°kG\C�.�U��QOX�h&\Cl.t R[hGk�qOQSl.t �_QSR[qSR[h ¼_l]t?XZQOiZi.R[h �Wn?qOq R[XZX.R[QSl.tGR[h<M�gI���.R[QzkkG\C��qSR[iZ��i.R{�G`CqOo�R[h?oNqOQSR�k 6= 0

MP1\ph?h�oNQOqOi!o�R[h&\pn���QSR �/R[QOa g�n?m_qOQSk�QS�.l]tGR[h°��qOo�`CXZQOiZt?aUnG��QOh

ZL

• zk | zn?hGk

zk | w

• t | z n?hGk t | w⇒ t | zk

• gI�WoNQO�?i A,B ∈ Z[i]¸?�.`�kG\C�]�

zk = A · z +B · w.

P1\C�Ig�qSR[abR[hFizkQS�Zi��?QS� \pn��2°n?qOiZQOY?qOQOmN\piZQS`Ch�a�QOi¦R[QOhGR[X¦gIQOh?tGR[QOi \pnG�

E(Z[i]) = {1, i,−1,−i} k�n?X.l.t�k�QSRR[X.�Zi.R[hy�/R[QSk?R[h�© R�k�QOh?oNn?h?o�R[h#� R��ZiZo�R[qSR[oNi@M�®¯QOX hGR[h?hGR[hzk

��R[QOhGR[h��¤oNX.V"�Ni.R[h�o�R[abR[QOhG�]\pabR[h�·JR[QOqSR[XW��`Chzn?hGk

wn?hGk��.l.t?X.R[QO�/R[h

zk = ggT (z, w).

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K&K ³ 6 ��� �����¾À 6 ���¹Ày�{�¾��� � �I�{���¦���

= 4 µ�³ � �GK ��&�(f ∈ N

&�(*5'& �� $N>�(?+ < & : ��( /N&�5," � $ 7 + 6�< #"�"

p | f2 + 1

� " +ggT (f + i, p) = a+ ib 6 " � /�(*>?+

p = a2 + b2.

! 8F79> } 798F; ³ � � Ã ¼_R[Qp = 13

n?hGkf = 720

M�gI�¦oNQOqOif 2 +1 = 13 · 39877 M�® QOX � ?t?X.R[h�k?R[h�g�n?m_qOQSk�QS�.l.tGR[h��qOo�`CXZQOiZt?aUnG��a�QOi

f + in?hGk

pk�n?X.l.t<M

720 + i = 55 · 13 + (5 + i)

13 = 2 · (5 + i) + (3− 2i)

5 + i = (1 + i) · (3− 2i) + 0

��qS�.`�QS�ZiggT (720 + i, 13) = 3− 2i

n?hGkyQOh�k?R[XW·+\pi13 = 32 + (−2)2.

DE&GF�&�(H" @ ® QOX�t&\p�/R[h a + ib | f + i¸a + ib | p n?hGk¾R[QOhGR�P \pX.�.i.R[qOqOn?h?o a + ib = A(f + i) + Bp

¸ \pqOqSR��QOh

Z[i]MG��nG�

a+ ib | p � `CqOoNia2 + b2 | p2 M?� Y?XZQS`CXZQ<oNQO�?iWR��Whjn?h�k�X.R[Q�°VCoNqOQSl]t?m�R[QOi.R[h<M� �

a2 + b2 = 1¸

© �a2 + b2 = p2 n?hGk � �

a2 + b2 = pM_® QOX �@R[QOo�R[h<¸GkG\C�.��k�QSR1�H�pqOqSR ���¤n?hGk�© �¤h?QSl.tFi!abVCoNqOQSl.t��.QOhGkĸ?�.`

kG\C�.� � � \pn���iZX.R[i.R[h�a�nG�.�@M� �J£ �Zi

a2+b2 = 1¸N�]` � `CqOoNi

a+ib ∈ {1, i,−1,−1, 1} = E(Z[i])MN�WnG��k?R[XIP1\pX.�Zi.R[qOqOn?h?o

a+ib = A(f+

i)+BpR[XZt&\pqOi.R[h{��QOX¤k�n?X.l]t�°n?qOiZQOY?qOQOmC\piZQS`Ch{a�QOi¤k?R[a £ hj��R[X.�.R[h�R[QOhGR!P1\pX.�Zi.R[qOqOn?h?o

1 = A′(f + i)+B′pM

°n?qOiZQOY?qOQOmC\piZQS`Ch°a�QOif − i R[XZoNQO�?iWkG\ph?h

f − i = A′(f2 + 1) +B′(f − i)p

P1\p | f2 + 1

¸G� ?X.k?RUhjn?h ��`CqOo�R[h<¸kG\C�.�p | f − i ¸G� \C��` ÄR[hG�ZQSl]tjiZqOQSl.tsh?QSl]tji���n?iZXZQ i@M�WqS�.`bQS�ZiW�&\pqOq+���

n?h?abVCoNqOQSl.t<M© ��gI�¤QS�Zi

(a+ ib) | p ¸jn?hGkb��QOX �.l.t?X.R[QO�/R[h (a+ ib)(c+ id) = pM!��QSR[X]\pnG� ��`CqOoNi

(a2 + b2)(c2 +d2) = p2 M£ �Zi

a2 +b2 = p2 ¸N�.`�� `CqOoNic2 +d2 = 1

n?hGkb�.`Ca�QOiIQS�.ic+ id ∈ E(Z[i])

n?hGkp | a+ ib

M���nG�a+ ib | f+ i

� `CqOoNi!kG\ph?h<¸&kG\C�.�!\pnGl]tp | f + i

¸�� \C��` /R[hG�.QSl.tjiZqOQSl]t°h?QSl]tji���n?iZXZQ i@M ½�gIP® QSR�¡GhGk?R[h���QOX!�.`CqSl]t�R[QOh

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³ �O³Z[i]

Á E ~ ÄÁ+� � 8FE � 0 E "�Á = ~ 2 = 4 8_E K/Ã

= 4 µ�³ � ���

• p ��: ( � ⇒ p | (p− 1)! + 1

• p ��: ( � ;5 < ;53/N& : < & ⇒ p | (p−12 )! + (−1)

p−1

2

DE&GF�&�(H" @ ®¯QOX.k��.Y��pi.R[X�h&\Cl.t?o�R[tG`CqOi@M ½�gIP

z{0&2p0 ;�;9= 2 ³ � �$� � " +p = 4k + 1 ��: ( � ;5 < f := ( p−1

2 )! 6 " � /�(*>?+

p | f2 + 1

= 4 µ�³ � � � ��(*5'& �� $N>n(H" + < : "G+L&�>*>21� : >H" � ��� & � 5 ��F�&�( � <�: +L&�5 6 F�&�5N5 7L8;: >*> & � : ( � � $N> &�5

< & : � � :�� 4k + 3 < & : &�5 + " ��: & �L$'&�5 < & � � � 5'&�5 +�(*5 < & : � : ( ��7 4�+ � : �;& : > &W/ ;53/ /N& : < & (H" + @6y8�E+7 4 7 0 E ³ � � 5 g�QOh��WQOh?o

RtGR[Q�Ni �*# ���X�"� ) ��� $�� � � �N¸_�¤R[h?h¾R[QOhGRc�W�?�?QOqSk�n?h?o

δ : R −→ N

R�^�QS�ZiZQSR[XZi!a�QOi!k?R[XWg�QOo�R[hG�]l.t&\ ��i@¸&kG\C�.� � ?XW\pqOqSRa, b ∈ R ¸ a, b, 6= 0

R[QOhqR�^�QS�ZiZQSR[XZi@¸G�.`�kG\C�]�

δ(a− qb) < δ(a).

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K$� ¶ � �I� " �����U� 6 ��� À�� � � �c� �����y� �������

� � .' � '���"' ')(�� (�� � �{'���"��'H��(/,.'

6y8�E+7 4 7 0 E¯¶ ��5 g�QOhGR 2 $� �'$ % � � � $��$# �V5D� � ��QS�Zi R[QOh��)\N\pX(M, ∗) �/R��Zi.R[tGR[hGk�\pnG� R[QOhGR[X6¾R[h?o�R M��nG�]\pa�abR[h�a�QOi�R[QOhGR[X � $��$# �V5D�$�� � � \pnGl.t�o�R[h&\ph?hFi�R[QOh � ��!+� ��# � �W\pn��

M

∗ : M ×M −→M,

�¤R[qSl]tGRc��� R[QDg�qSR[abR[hFi.R[ha, b ∈M R[QOh��¤R[QOi.R[X.R���g�qSR[abR[hji

a ∗ b ∈M ��nG`CX.k�hGR[i@M

! 8F79> } 798F;98¾¶ � � © R[QS�ZY?QSR[qSR1��`Ch�¥IR[XZm_h! ?Y���n?h?o�R[h�L

• (N,+)¸(N, ·) ¸ (Z,+)

¸(Z, ·) ¸ (Q,+)

¸(Q, ·) ¸ (R,+)

¸(R, ·) ¸ (C,+)

¸(C, ·)

• (R, ∗) ¸G� `C� R[Q a ∗ b := max(a, b)

£ �ZiM = {m1,m2, . . . ,mr}

R[QOhGR{R[hGk�qOQSl]tGRE°R[h?o�RN¸+�.`°m�VCh?hGR[h ��QOX�k�QSR�£�h�� `CXZa�\piZQS`Ch ?� R[X�k�QSR{¥IR[X#�m_h! ?Y���n?h?o�QOhyR[QOhGR[X � $��$# �V5D� � �()�� ��! $ � �($ � R��ZiZt&\pqOi.R[h<M�P�QSR�� QS�Zi R[QOhGR

r× r ·D\p�/R[qOqSRN¸_�¤`C�/R[QJ\phyk?R[X!¼Fi.R[qOqSR(i, j)

kG\C�Wg�XZo�R[�?h?QS�mi ∗mj

R[QOh?o�R[iZX]\po�R[h���QOX.kÄM

∗ m1 m2 . . . mn

m1 m1 ∗m1 m1 ∗m2 . . . m1 ∗mnMMM MMM MMM MMMmn mn ∗m1 mn ∗m2 . . . mn ∗mn

6y8�E+7 4 7 0 E¯¶ � � g�QOhGR*¾R[h?o�RMa�QOiDR[QOhGR[X<¥IR[XZm_h! ?Y���n?h?o ∗ tGR[Q�Ni�� �"� ! �"� �+5�5 $[¸p�¤R[h?h�k�QSR¦¥IR[XZmFh ?Y��9n?h?o\C�.�.`C��Q�\piZQO��QS�Zi�L

a ∗ (b ∗ c) = (a ∗ b) ∗ c

6y8�E+7 4 7 0 E¯¶ �O³ g�QOhGR � \pqO�?oNXZn?Y?Y R1tGR[Q�Ni 2 !" !"�X�N¸�� R[h?h°R��!R[QOh�g�qSR[abR[hFie ∈M oNQO�?i@¸G�]`�kG\C�.�

e ∗ a = a = a ∗ e

� ?X \pqOqSRa ∈ M M

etGR[Q�Ni1kG\ph?h $ �������"�($$) �($ % $� �+k?R�� °`ChG`CQSk?�@M

b ∈ M tGR[Q�Ni � � $��$) $$) �($ % $� � � �a ∈M ¸?� R[h?h

a ∗ b = e = b ∗ a.

! 8F79> } 798F;�¶ ��¶ ¼_R[QM

k�QSR%°R[h?o�R�k?R[Xc���?�?QOqSk�n?h?o�R[h {0, 1} −→ {0, 1} MJP1\���`ChºoNQO�?iUR�� ��¼Fi Gl]ml�Ä��QOXhGR[h?hGR[h��ZQSRa¸b¸c¸dL

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K��

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

b dca

��qS�.`

a(0) = 0, a(1) = 1;

b(0) = 1, b(1) = 0;

c(0) = 0, c(1) = 0;

d(0) = 1, d(1) = 1.

��n��M

k?R�¡Gh?QSR[X.R[h°��QOX!R[QOhGR1¥IR[XZm_h! ?Y���n?h?o ◦ k�n?X.l.t �WQOhFi.R[X.R[QOh&\phGk?R[X]\pnG��� ?t?XZn?h?o����¤R[QSR[X��W�?�?QOqSk�n?h?o�R[h<MP�QSR ¥IR[XZmFh ?Y��9n?h?o��.i]\p� R[qOqSR � ?X ◦ \pn�� M QS�Zi!kG\ph?h

◦ a b c d

a a b c d

b b a d c

c c c c c

d d d d d

= 4 µ°¶ �SK � 5 &�(*5'&���5�( < /�(213+ &."0/N&�5 8_7�E 5'&�N+ : > &." ��> & � &�5 + @ ��(*5���> & � &�5�+�(*5 &�(*5'&

���5�( <

1,&."W(?+���+ $ � �L$," +L&�5," 8_7�E (*5 #& : " &." ��> & � &�5�+ @DE&GF�&�(H" @ ¼FQOhGk e1 n?hGk e2 ���¤R[Q<hGR[n?iZX]\pqSR�g�qSR[abR[hji.RN¸�.` � `CqOoNi

e1 = e1 ∗ e2 = e2.

¼_R[QSR[hb1n?hGk

b2QOhj��R[X.�W��n

aM?P1\ph?h�oNQOqOi�L

b1 = b1 ∗ e = b1 ∗ (a ∗ b2) = (b1 ∗ a) ∗ b2 = e ∗ b2 = b2.

½�gIP®�R[h?h

aR[QOh�£�hF��R[X.�.R���t&\pi@¸GkG\ph?h�� R[�@R[QSl]t?hGR[h���QOXWkG\C�!R[QOhGk?R[n?iZQOo{� R��.iZQOa�aUi.Rc£ hj��R[X.�.Rc��n

aa�QOi

a−1 M

6y8�E+7 4 7 0 E¯¶ �9à g�QOh�¾`ChG`CQSkytGR[Q�Ni C � �+5�5 $[¸?�¤R[h?h��#R�k?R��Wg�qSR[abR[hji�R[QOh�£�hF��R[X.�.R���t&\pi@M ����QSR[X�QS�Zi�hG`Fl]t�R[QOh?a�\pqDk�QSR1P R�¡Gh?QOiZQS`Ch°��nG�]\pa�abR[h?o�R��.l.t?XZQSR[�/R[h�LJ � ( <MY;:� J$K �#OLI-& � � : 9;<M:O> : JLK6S P � :OSXSX:OI�:=<MY;:� I�<?@X@�:=K O$I-& � � :+&DY�� :=<MY �BP Y P <)� :=<MY;:���,":O< �+I�<?@X@�:=K O$I-& � � :�U

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Ã�� ¶ � �I� " �����U� 6 ��� À�� � � �c� �����y� �������

6y8�E+7 4 7 0 E¯¶ � � g�QOhGRa¾R[h?o�RG��nG�²\pa�abR[h�a�QOi¦R[QOhGR[X¤¥IR[XZmFh ?Y��9n?h?o ∗ : G×G −→ G

tGR[Q�Ni C �$�V5�5 $�¸�¤R[h?h ��`CqOo�R[hGk?RUk�X.R[QD��XZn?Y?Y R[h&\�^�QS`CabRUR[X�� ?qOqOiW�ZQOhGkÄM

G1) : ∀a, b, c ∈ G : (a ∗ b) ∗ c = a ∗ (b ∗ c)G2) : ∃e ∈M ∀a ∈ G : a ∗ e = e ∗ aG3) : ∀a ∈M ∃b ∈ G : a ∗ b = e = b ∗ a

P1\C� QOh��1e �¤o�R ��`CX.k?R[XZi.R1gIqSR[abR[hji!QS�Zi�R[QOhGk?R[n?iZQOobn?hGk�tGR[Q�Ni $ �������"�($$) �($ % $� � M�P1\C� QOhG3)

o�R ��`CX.k?R[XZi.Rg�qSR[abR[hFi

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an?hGky��QOX.kya�QOi

a−1 hG`CiZQSR[XZi@M

6y8�E+7 4 7 0 E¯¶ � � ®�R[h?h � ?X���� R[QDg�qSR[abR[hFi.Ra, b ∈ G oNQOqOi

a ∗ b = b ∗ a,

�.`b�²\po�R[h���QOX�La#&!(%'%'����� $�� ����kÄM t<M?��R[XZi]\pnG�.l]tji �Wa�QOi

bMG·JXZQ i�k�QSR���\pn��I\pqOqSRU�)\N\pX.Rc��`Ch°gIqSR[abR[hji.R[h¾��n<¸

�.`��]\po�R[h���QOX�LGQS�Zi #&!(%'%'� � ����� � `Fk?R[XW\pnGl]t�L G QS�Zi ��! $�� ) � �FM � �

! 8F79> } 798F;98¾¶ �95�� ® QOX�t&\p� R[h°�.l.tG`Ch�R[QOh?QOo�RU© R[QS�.Y?QSR[qSR1��`Ch°\p�/R[qS�.l.tGR[h¾� XZn?Y?Y R[h�o�R��]R[tGR[h<M

• ¼_R[Q R R[QOh�� R[qOQSR[�?QOo�R[X!�WQOh?oG¸Fn?hGk���R[XZo�R��.�]R[hy��QOX k�QSR °n?qOiZQOY?qOQOmC\piZQS`Ch<M?P1\ph?hyQS�Zi(R,+)

R[QOhGR \p�/R[q ��]l.tGR���XZn?Y?Y/R����W^_QS`CabR1��L�������� � ��M

• ¼_R[Q R R[QOhº�/R[qOQSR[�?QOo�R[XU�!QOh?oG¸E(R)

k�QSR%¾R[h?o�R�k?R[Xcg�QOh?tGR[QOi.R[h ��`ChRM/P1\ph?h�QS�Zi

(E(R), ·) R[QOhGR\p�/R[qS�.l]tGR¾��XZn?Y?Y/RNMI£ hG�Z� R��]`ChGk?R[X.Ry�?QOqSk?R[h k�QSR�gIqSR[abR[hji.R 6= 0QOh R[QOhGR[a TWVCXZY/R[X�R[QOhGR°\p�/R[qS�.l]tGR

� XZn?Y?Y R1�/R[� ?oNqOQSl.t · M

• P�QSR�¾R[h?o�Rµn := {z ∈ C | zn = 1} k?R[X n � i.R[h g�QOh?tGR[QOi.�Z��n?XZ�@R[qOh a�QOi�°n?qOiZQOY?qOQOmN\piZQS`Ch \pqS�

¥IR[XZm_h! ?Y���n?h?o�QS�.i!R[QOhGRU\p� R[qS�]l.tGR���XZn?Y?Y/RNM

� ��� ����� �'����� ��� ��� ����� �#� � ��� ���®�R[h?h°Æ����ZR[mFi.R

a, b, c, . . . , d\pn��+R[QOhGRc\phGk?R[X.Rc®�R[QS�.R1\ph?o�R�`CX.k�hGR[i!�¤R[X.k?R[h<¸?��n?a © R[QS�ZY?QSR[q

c, a, d, . . . , b¸

kG\ph?h��]\poNi�a�\ph<¸?kG\C�]�!k�QSRUÆ����#R[m_i.R 5 $�� %'����� $�� �Ä��n?X.k?R[h<MVs\piZtGR[a�\piZQS�.l.t°QS�Zi�k�QSR��W\pn�����n�� \C�.�]R[h�\pqS�!R[QOhGR���?�?QOqSk�n?h?oG¸&� R[qSl.tGR �#R�k?R[a Æ����ZR[mFi�kG\C��\ph¾�.R[QOhGR�¼Fi.R[qOqSR�iZX.R[i.R[hGk?R�Æ����ZR[m_iW��nG`CX.k�hGR[i@¸Ä\pqS�.`

a 7→ c, b 7→a, c 7→ d, . . . , d 7→ b

M

6y8�E+7 4 7 0 E¯¶ ��55 ¼_R[QM

R[QOhGR1� R[qOQSR[�?QOo�R�°R[h?o�RNMJ � <M:=KMS :=YDI�<�� � � :OK P ∗ �!��Y�" P † �!��"ZS

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¶ ��5 6�798 � 8 2C�sÁ<4 = 4 7 0 EJ>@´ 2pÁD}+} 8 à 5

• g�QOhGR � $��$%'� � ������!" �9��`ChM

�DQS�.i)R[QOhGR © Q �ZR[mFiZQS`Chσ : M −→M

MCP�QSR6°R[h?o�R!\pqOqSR[XI�+R[XZaUn?i]\piZQS`ChGR[h��`Ch

MhG`CiZQSR[X.R[h���QOX

S(M)M

• ¼FQOhGk σ, τ ∈ S(M)¸J�]`yQS�Zi�\pnGl]t k�QSR{TW`Ca�Y/`N�ZQOiZQS`Ch ��`Ch �W�?�?QOqSk�n?h?o�R[h

σ ◦ τ : M −→ MR[QOhGR

�+R[XZaUn?i]\piZQS`Chsn?hGky��QOX.k�\pnGl]t � ��!+�(�*# �J��`Chσn?hGk

τo�R[h&\ph?hji@M

= 4 µ°¶ ��5 � �B( & � &�53/N&S(M) �

(?+ < & : � ��� � � "�(?+ ( � 5 ◦ >H" �'& : 4 5 8 ��7 ;53/ (H" + &�(*5'& �0: � � & @6y8�E+7 4 7 0 E¯¶ ��5 �

Sn := S({1, 2, . . . , n}) MG£ �Zi σ ∈ Sn ¸&�.`��.l.t?X.R[QO�?iWa�\ph°\pnGl.t

σ =:

(

1 . . . n

σ(1) . . . σ(n)

)

.

! 8F79> } 798F;98¾¶ �95_³ � ?Xn = 2

oNQO�?iWR������ R[QJ�+R[XZaUn?i]\piZQS`ChGR[h

e :=

(

1 2

1 2

)

, α :=

(

1 2

2 1

)

,

n?hGk�k�QSR ¥IR[XZm_h! ?Y���n?h?o��Zi]\p�/R[qOqSRUQS�Zi◦ e α

e e α

α α e� ?X

n = 3oNQO�?iWR��

6�+R[XZaUn?i]\piZQS`ChGR[h

e :=

(

1 2 3

1 2 3

)

, a :=

(

1 2 3

2 3 1

)

, b :=

(

1 2 3

3 1 2

)

,

α :=

(

1 2 3

1 3 2

)

, β :=

(

1 2 3

3 2 1

)

, γ :=

(

1 2 3

2 1 3

)

,

n?hGk�k�QSR ¥IR[XZm_h! ?Y���n?h?o��Zi]\p�/R[qOqSRUQS�Zi

◦ e a b α β γ

e e a b α β γ

a a b e γ α β

b b e a β γ α

α α β γ e a b

β β γ α b e a

γ γ α β a b e

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à � ¶ � �I� " �����U� 6 ��� À�� � � �c� �����y� �������

= 4 µ°¶ ��5�¶ �B( & � : � � & Sn $ + /N&�5 n!��> &�&�5 +L& @

6y8�E+7 4 7 0 E¯¶ ��5�K ®�R[h?hn1, . . . , nk

o�R[h&\pnk��R[X.�.l.t?QSR�k?R[hGR�g�qSR[abR[hFi.R���`Ch {1, . . . , n} �ZQOhGkĸ k ≥ 2

¸kG\ph?h��/R[�@R[QSl.t?hGR[h°��QOX!a�QOi

(n1 . . . nk)k�QSRU�DR[XZa�n?i]\piZQS`Ch

(n1 . . . nk) : {1, . . . , n} −→ {1, . . . , n}ni 7→ n(i+1 � �

�k)

� ?Xi = 1, . . . , k,

m 7→ m� ?X

m /∈ {n1, . . . , nk}.

P�QSR1�DR[XZa�n?i]\piZQS`ChGR[h(n1 . . . nk)

tGR[Q��R[h %�� #"$ � �'$���� �" �'$kM

r��_m�R[q&k?R[X¤�8�ph?o�RW���¤R[Q tGR[Q��R[h�\pnGl]t � ���"�)�5 !;)�� ����!"#$ &M�·<X]\phG�.Y `N�ZQOiZQS`ChGR[h��ZQOhGk�\pqS�]`c�DR[XZa�n?i]\piZQS`ChGR[h<¸?k�QSR���¤R[QDg�qSR[abR[hji.R���R[XZi]\pnG�.l.tGR[h°n?hGk��.`ChG�Zi�h?QSl]tji.��a�\Cl.tGR[h<Mr�¤R[Q)r��Fm�R[q

(n1, n2, . . . , nk)n?hGk

(m1,m2, . . . ,ml)tGR[Q��R[h �"� ) � � #&� ¸��¤R[h?h

{n1, n2, . . . , nk} ∩ {m1,m2, . . . ,ml} = ∅.

! 8F79> } 798F;98¾¶ �95&Ã

(123) =

(

1 2 3

2 3 1

)

QS�Zi!R[QOh¾r��_m�R[qJk?R[X!�8�ph?o�RUk�X.R[Q M

(12) =

(

1 2 3

2 1 3

)

QS�Zi!R[QOh¾r��_m�R[qJk?R[X!�8�ph?o�Rc��� R[Q M

= 4 µ°¶ ��5�� & < & ��& :�� N+2 + ( � 5 4� 5N5 /N&." � $ : ( & 1,&�5 F�& : < &�5� >H" � : � < ;4'+ � 5 � : F�&�(H" & < (H" � ;5N4'+L&�5���N43&�>*5 < & : � � 53/N& > 1 @ �E( &." & � � $ : &�(21�F�&�(H" &=(H" + 1#(H"E 7 � &�(*$'&�5 7 � > /N& &�(*5 < &�N+ ( / @! 8F79> } 798F;�¶ ��5��

σ =

(

1 2 3 4 5 6 7 8 9

5 1 7 4 2 9 8 6 3

)

P1\ph?h�QS�Ziσ = (9 6 8 7 3)(2 5 1)

� ��� �'��� ��� � � � ����� � �� �����6y8�E+7 4 7 0 E¯¶ � �$� ¼_R[Q

GR[QOhGR���XZn?Y?Y/RN¸

g ∈ G M&P�QSR � !�� $� � $�gn¸n ∈ Z

¸?�¤R[X.k?R[hsX.R[m_n?X.�ZQO� ��R��ZiZo�R[qOoNik�n?X.l.t

gn = g ∗ gn−1,� ?X

n ∈ N, g0 = e.

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¶ � � ��0G4 8_EJµj8FE Á E ~ ��2 ~ E Á EJ´ à �

� ?Xn < 0

�.R[iZ�@R[h���QOXgn = (g−n)−1

gI��oNQOqOi�kG\ph?h ���Wn���oj\p�/R � �gn ∗ gm = gn+m ∀n,m ∈ Z.

! 8 � 8 2 : Á ED´ ¶ � �Ä5(gh)2 = ghgh

¸/n?hGk¾k�QSR���QS�Zi�QOa �WqOqOo�R[abR[QOhGR[h�R[i � \C�U\phGk?R[X.R��1\pqS�g2h2 = gghh

MTW`Ca�a�n?iZQSR[X.R[h

gn?hGk

h¸GkG\C�!tGR[Q�Ni

gh = hg¸G�.` � `CqOoNi!a�QOiW£ hGk�n?mFiZQS`ChyqSR[QSl]tji@¸ kG\C�.�

(gh)n = gnhn.

6y8�E+7 4 7 0 E¯¶ � � � ¼_R[QGR[QOhGR�� XZn?Y?Y R1n?hGk

a ∈ G M?P�QSR�� �B�"��� ����`Ch a QS�Zi!k?R�¡Gh?QSR[XZi�\pqS�

ord(a) :=

min{n ∈ N; an = e}, � \pqOqS� ∃n ∈ N : an = e,

∞ �.`ChG�Zi.

! 8F79> } 798F;98¾¶ � � �

• G = Ra�QOi

+\pqS����XZn?Y?Y/R[hG`CY R[X]\piZQS`Ch<M�WqOqSRUg�qSR[abR[hji.R

x 6= 0t&\p�/R[h�n?hGR[hGk�qOQSl.tGR��X.k�hjn?h?oGM

• G = µ8¸<k�QSR�� XZn?Y?Y Rbk?R[X

8� i.R[h g�QOh?tGR[QOi.�Z��n?XZ�@R[qOh<¸Da�QOi��n?qOiZQOY?qOQOmN\piZQS`Ch m�`Ca�Y?qSR�^?R[X�r/\pt?qSR[h \pqS�

� XZn?Y?Y R[hG`CY/R[X]\piZQS`Ch<MG® QOX�¡GhGk?R[h

ord(1 + i

2) = 8, ord(i) = 4, ord(−1) = 2

• ¼_R[Q z = (n1, n2, . . . , nk) ∈ SnR[QOh°r��Fm�R[qJk?R[X��8�ph?o�R

kMÆ ÄR[hG�ZQSl.tFiZqOQSl.t¾QS�Zi

ord(z) = k.

• ord((123) ◦ (45)) = 6

• ord((1234) ◦ (5678910)) = 12

= 4 µ°¶ � �?³ � "G+σ ∈ Sn

� : � < ;4'+ � 5 < (H" � ;5N4'+L&�5 ���343&�>*5 z1, z2, . . . , zr � (?+ ord(zi) = ki 6 " � (H" +

ord(z) = kgV (k1, k2, . . . , kr).

! 8F79> } 798F;�¶ � �&¶ P1\C�!g�qSR[abR[hji(12345)(6789) ∈ S9

t&\pi��X.k�hFn?h?o20M

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Ã_³ ¶ � �I� " �����U� 6 ��� À�� � � �c� �����y� �������

= 4 µ°¶ � �GK ��&�(G

&�(*5'& 1,&�>*( & 1#( /N& � : � � & @� ( � � " + x ∈ G 6 " � /�(*>?+xn = e ⇔ ord(x) | n

� (*( � � (*5 < x, y ∈ G 4�����

N+ ( & : &�5 < & ��> & � &�5 +L&B" � /�(*>?+

ord(xy) | kgV (ord(x), ord(y))

� (*(*( � � " +� ;)3& : < & � ggT (ord(x), ord(y)) = 1 6 " � /�(*>?+

ord(xy) = ord(x) · ord(y) (= kgV (ord(x), ord(y))

DE&GF�&�(H" @ �9Q ��¼_R[Q k = ord(x)M)£ �Zi

n = q · k ¸)�.`°QS�Zi�` /R[hG�.QSl.tjiZqOQSl]t xqk = (xk)q = eq = eM+� ?X�k�QSR

Ç�a�m�R[t?XZn?h?o��/R[hjn?iZ�@R[h¾��QOXW·JR[QOqOn?h?o�a�QOiW�WR��.i!n?hGk��.l]t?X.R[QO� R[hn = q · k + r

¸?a�QOi0 ≤ r < k

MGgI� ��`CqOoNie = xn = xqk+r = xqkxr = xr

M?g¦� ��`CqOoNir = 0

¸&\pqS�.`k | n M

�9QOQ ��g¦��oNQOqOi(xy)m = xmym

¸I� R[h?hxy = yx

M¦£ �ZimR[QOh¯¥�QSR[q� \Cl.tGR�����`Ch

ord(x)n?hGk

ord(y)¸ �.`

� `CqOoNi(xy)m = 1

M+£ hG�Z� R��]`ChGk?R[X.R{oNQOqOi�k�QSR�� � ?Xm = kgV (ord(x), ord(y))

M���nG�(i)

� `CqOoNi�hFn?h<¸�kG\C�.�ord(xy) | kgV (ord(x), ord(y))

M�9QOQOQ ��¼_R[iZ�@R

a = ord(x)n?hGk

b = ord(y)M £ �.i

(xy)n = e¸ �.` � `CqOoNi{iZXZQO�_Q�\pqSR[XZ�¤R[QS�]R

e = (xy)an =

xanyan = yanMA°QOi

(i)��`CqOoNi

b | an MÄP1\ ggT (a, b) = 1¸ ��`CqOoNi1hFn?h

b | n M<¼_`�\pnGl]t a | n n?hGkº�.`Ca�QOia · b = kgV (a · b) | n M ½�gIP

� ��� � ��� ����� � ��� ��� �6y8�E+7 4 7 0 E¯¶ � �/à ¼_R[Q

(G, ∗) R[QOhGR�� XZn?Y?Y R n?hGk U ⊂ G R[QOhGRc·DR[QOqOabR[h?o�RNM ��QOqOiLNMe ∈ U ¸

e�Ma, b ∈ U ⇒ a ∗ b ∈ U ¸

w�Ma ∈ U ⇒ a−1 ∈ U ¸

�.`�QS�Zi(U, ∗) R[QOhGR���XZn?Y?Y/R n?hGkytGR[Q�Ni � � $�� ���$�V5�5 $W��`Ch

(G, ∗) M

! 8F79> } 798F;98¾¶ � �$�

• nZ := {n · z, z ∈ Z} ⊂ ZQS�Zi!R[QOhGRcÇ�hji.R[XZoNXZn?Y?Y/R�� � ?X

n ∈ Z��M

• µn ⊂ C∗ = C \ {0} �9a�QOi�°n?qOiZQOY?qOQOmC\piZQS`Ch�m�`Ca�Y?qSR�^�R[X�r/\pt?qSR[h°\pqS� ¥IR[XZm_h! ?Y���n?h?o � QS�Zi�R[QOhGR Ç�hji.R[X#�oNXZn?Y?Y/RNM

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¶ � � � E 4 8 2 ´ 2pÁD}+} 8FE ÃG¶

• ��R[t?abR[h���QOXWR[QOh���R�l.tFi.R�l.m�\pnG�!Tc\pXZi.`Ch�n?hGk�hjn?a�abR[XZQSR[X.R[h¾��QOX!k�QSRcgIl.m_Y?n?h?m_i.R1¸2¸3¸4M

1 2

4 3

2 1

3 4

4 3

1 2

3 4

2 1(12)(34) (14)(23) (13)(24)

�jR�k?R[X�P�X.R[tjn?h?o�k?R����WR�l]tji.R�l.m��!qOQSR � R[XZi!n?hG�!R[QOhGRU�DR[XZa�n?i]\piZQS`Ch¾k?R[XWg¦l.m_Y?n?h?mFi.RNM

• ®�R[qSl]tGRU�+R[XZaUn?i]\piZQS`ChGR[hºR[XZt��pqOi!a�\ph��/R[QOa P�X.R[tGR[h�R[QOhGR���® ?X���R[qS��±

• £ �Zi {Hi}i∈IR[QOhGRº�/R[qOQSR[�?QOo�RºTW`CqOqSR[m_iZQS`Chu��`Ch Ç�hji.R[XZoNXZn?Y?Y/R[hu��`Ch

G¸��.` QS�Zi�k?R[Xyo�R[abR[QOhG�]\pabR

P�n?X.l]tG�.l.t?h?QOiZi ∩i∈IHi��QSR�k?R[X�R[QOhGR1Ç�hji.R[XZoNXZn?Y?Y/Rc��`Ch

GM

6y8�E+7 4 7 0 E¯¶ � �$� ¼_R[QGR[QOhGR�� XZn?Y?Y RN¸

S ⊂ G R[QOhGR �/R[qOQSR[�?QOo�RU·JR[QOqOabR[h?o�RU��`ChGM

< S >:=m_qSR[QOhG�Zi.Rc�hji.R[XZoNXZn?Y?Y/Rc��`Ch

G¸G�¤R[qSl.tGR�\pqOqSRcg�qSR[abR[hFi.RU��`Ch

SR[hFiZt��pqOi

® QOXU�]\po�R[h�LG��QOX.k $ � � $ � � �¦��`Ch

S¸/�¤R[h?h

< S >= GM<£ �Zi

S = {g1, g2, . . . , gr}¸J�.`yq�\C�.�.R[hº��QOX�k�QSR

T q�\pa�abR[XW� R[obn?hGk��.l]t?X.R[QO� R[h< S >=< g1, . . . , gr > .

£ �ZiG =< g >

¸?�]`�tGR[Q�NiGR[QOhGR � � # ��� ) ��� $ C � �V5�5 $[¸&n?hGk�R[QOh��.`CqSl.tGR��

gtGR[Q�Ni ��� $ � �'$ ����`Ch

GM

! 8F79> } 798F;98¾¶ � � �

• £ �Zi ord(g) = n¸?�.`�oNQOqOi

gk = gk+q·n� ?XW\pqOqSR

q ∈ ZM&¼_`Ca�QOi!QS�Zi

< g >= {1, g, g2 , . . . , gn−1}.

��qS�.` Lord(g) = | < g > |

• P�QSR�r/\pt?q ζ = cos(2πi/n) + i sin(2πi/n)QS�Zi!R[QOh�g�XZ�@R[n?o�R[X � ?X

µnM

µn =< ζ >

• Z =< 1 >=< −1 >¸nZ =< n >=< −n > M

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Ã�K ¶ � �I� " �����U� 6 ��� À�� � � �c� �����y� �������

• < (123) >= {e, (123), (132)} ⊂ S3M

• < (12), (123) >= S3MS3

mC\ph?h¾h?QSl.tFick�n?X.l.t�R[QOhsg�qSR[abR[hjicR[XZ�@R[n?oNi � R[X.k?R[h<¸/�¤R[QOq�R���QOhS3

m�R[QOhgIqSR[abR[hji k?R[X�Æ�X.k�hjn?h?o

6oNQO�?i@M

S3QS�Zi�m�R[QOhGR1� �_mFqOQS�]l.tGR���XZn?Y?Y/RNM

6y8�E+7 4 7 0 E¯¶ � � 5 gI�!�]R[QSR[h(G1, ∗1)

n?hGk(G2, ∗2)

���¤R[Q+��XZn?Y?Y/R[h<MGg�QOhGRU�W�?�?QOqSk�n?h?o

ϕ : G1 −→ G2

tGR[Q�Ni � !(% !(% !"��5 �'�7) %'��) �9��`Ch¾��XZn?Y?Y/R[h���¸?�¤R[h?h � ?XW\pqOqSRa, b ∈ G1

ϕ(a ∗1 b) = ϕ(a) ∗2 ϕ(b)

oNQOqOi@MG£ �Ziϕ�?Q �#R[m_iZQO�/¸?�.`�QS�Zi!k�QSR1QOhF��R[X.�.RU���?�?QOqSk�n?h?o�R[�/R[h�� \pqOqS��R[QOh ��`Cab`Cab`CXZY?t?QS�.aUnG� ���Wn���oj\p� R � ��¸Gn?hGk

a�\ph�hGR[h?hFiϕR[QOhGR[h � ) !(% !"��5 �'� )$%'�*)¤��`Ch°��XZn?Y?Y/R[h<M

= 4 µ°¶ � � � ��&�(ϕ : G1 −→ G2

&�(*5 � : � � &�5N$ ��������� :�� $N(H" � " @ �! 5 "W(*5 <

ker(ϕ) := {g ∈ G1 | ϕ(g) = e2} ⊂ G1

;5 <Bild(ϕ) := {ϕ(g) | g ∈ G1} ⊂ G2

��5 +L& : / : � � &�5 @! 8F79> } 798F;98¾¶ � ��

• ¼_R[Q g ∈ G M?P R�¡Gh?QSR[X.Rϕ : Z −→ G, k 7→ gk.

g¦�!QS�Ziϕ(k + l) = gk+l = gkgl = ϕ(k)ϕ(l),

\pqS�]`�QS�ZiϕR[QOh ��`Cab`Cab`CXZY?t?QS�.aUnG����`Ch¾��XZn?Y?Y/R[h<MGgI��QS�Zi

Bild(ϕ) =< g > .

£ �Ziord(g) = n <∞ ¸?�.`�QS�.i ker(ϕ) = nZ

�_QS�Ziord(ϕ) =∞ ¸?�.`�QS�Zi ker(φ) = {0} M

• µ3 −→ < (123) > ⊂ S3k?R�¡Gh?QSR[XZiWk�n?X.l.t

ω 7→ (123)¸ω2 7→ (132)

¸GQS�Zi!R[QOh�£ �.`Cab`CXZY?t?QS�Za�nG�!��`Ch� XZn?Y?Y R[h<M

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¶ � � � E 4 8 2 ´ 2pÁD}+} 8FE ÃÃ

• µ4n?hGk

< (12)(34), (13)(24) > ⊂ S4�ZQOhGk h?QSl.tFi�QS�.`Cab`CXZY?t<¸!`C�F�¤`Ct?q�� R[QSk?R � XZn?Y?Y R[h �_QSR[X

gIqSR[abR[hji.R�t&\p� R[h<M = 4 µ°¶ � � ³ � = 4 µ � 0 E c � � bj� � �$� � & < & � : � � & (H" +%(H" ����� :�� $�� &�(*5'& : ��5 +L& : / : � � & &�(*5'& : ��& : ��N+2 + (

�5,"W/ : � � & @ ��(*5'& � : � � & � (?+ n ��> & � &�5�+L&�5 (H" +�(H"

����� :�� $ � &�(*5'& : ��5 +L& : / : � � & � 5 Sn @DE&GF�&�(H" @ ¼_R[Q G R[QOhGR���XZn?Y?Y/RN¸/n?hGkº�.R[Q

S(G)k�QSR���XZn?Y?Y/R�k?R[Xc�DR[XZa�n?i]\piZQS`ChGR[h k?R[Xcg�qSR[abR[hFi.R���`Ch

GM

� ?Xg ∈ G �]R[Q

Lg : G −→ G, a 7→ g · ak�QSRc�ÄQOh?m��Za�n?qOiZQOY?qOQOmC\piZQS`Ch°a�QOi

gMG© R[abR[XZm�RN¸&kG\C�.�

Lg·h(a) = (g · h) · a = Lg(Lh(a))¸G�]`�kG\C�.�

Lg·h = Lg ◦ Lh.£ hG�Z� R��]`ChGk?R[X.RbQS�Zi

(Lg)−1 = Lg−1

¸J�.`°kG\C�.�ci]\pi.��Cl]t?qOQSl.tLg ∈ S(G)

QS�Zi@MJ£ hG�Zo�R��]\pa�i�R[XZt&\pqOi.R[h ��QOX��.`R[QOhGRc���?�?QOqSk�n?h?o

L : G −→ S(G), g 7→ Lg.

LQS�.icR[QOh ��`Cab`Cab`CXZY?t?QS�ZaUnG�@¸J�¤R[QOq

Lg·h = Lg ◦ LhMLQS�Zi�\pnGl.t QOh �ZR[m_iZQO�lLÄQS�.i1h��pa�qOQSl.t

Lg = Lh¸<�.`

� `CqOoNig · a = h · h � ?X \pqOqSR

a ∈ G ¸QOhG�Z� R��]`ChGk?R[X.R � ?X a = eM&� k?R�¡Gh?QSR[XZi �.`Ca�QOi�R[QOhGR[h¾£ �.`Cab`CXZY?t?QS�.aUnG�

��`ChG\pn��

L(G) ⊂ S(G)M ½�gIP

! 8 � 8 2 : Á ED´ ¶ � � ¶ £ �Zi |G| = n¸G = {g1, g2, . . . , gn}

n?hGkg ∈ G

¸��.`sQS�Zig · gi = gσ(i)

� ?XbR[QOhσ(i) ∈ {1, 2, . . . , n} M/P1\ g · gi = g · gj

hjn?Xc\pn���iZX.R[i.R[hsmC\ph?h<¸�¤R[h?hi = j

¸&QS�Ziσ : i 7→ σ(i)

QOh �ZR[mFiZQO�/¸\pqS�.`�\pnGl]t��?Q �ZR[mFiZQO�/¸

σ ∈ Sn��hGk?R[X.�co�R��]\poNi�L8�n?qOiZQOY?qOQOmN\piZQS`Ch a�QOi

g ∈ G �/R[��QOXZmFi�R[QOhGRb�DR[XZa�n?i]\piZQS`Chk?R[X�gIqSR[abR[hji.R���`Ch

GM+P�QSR{rnG`CX.k�hjn?h?o

g 7→ σQS�Zi�kG\ph?h R[QOh QOh �#R[m_iZQO��R[X���XZn?Y?Y/R[h?tG`Cab`Cab`CXZY?t?QS�Za�nG�

G −→ SnM

= 4 µ°¶ � � K ��&�(G

&�(*5'&!&�5 < >*(��L$'& � : � � & ;5 < g ∈ G @ �! 5N5 /�(*>?+ord(g) | |G| ;5 <

g|G| = e.

DE&GF�&�(H" @ � � ?X G \p�/R[qS�.l]t���M¼_R[QG = {g1, g2, . . . , gn}

MGg¦��QS�Zi

{g1, g2, . . . , gn} = {gg1, gg2, . . . , ggn}.P�n?X.l.t�°n?qOiZQOY?qOQOmN\piZQS`Ch°k�QSR��.R[XWgIqSR[abR[hji.RUR[XZt&\pqOi.R[h°��QOX�L

n∏

i=1

gi =

n∏

i=1

(ggi) = gnn∏

i=1

gi

��qS�.`gn = e

M ½�gIPJ # � IX@�9 &DI�V"J M KM: M P ∗ �$��"�� P † �$��S�Q

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Ã�� ¶ � �I� " �����U� 6 ��� À�� � � �c� �����y� �������

� ��� � ��� � � ��� � � � ��� � � � ����� �'����� ��� � ��������&� S `<�Vb ] � � U � � R[X�� \phGk�n?a 1870

kG\C�W\pnGl.t��#R[iZ��i�hG`_l.t�Y `CY?n?q �pX.R�¼_l]t?QSR[� R[Y?n?����qSRNM g�X � `CX.k?R[XZi.Rc�.R[QOhGR�<R��.R[X�\pn��Z¸_\pnG�Zo�R[tGR[hGk���`Chyk?R[X qOQOh?m�R[h��+`N�ZQOiZQS`Chyk�n?X.l.t�¥IR[X.�.l]t?QSR[� R[h�k�QSR�X.R�l.tFi.R1�D`N�.QOiZQS`Ch���n�R[XZX.R[QSl.tGR[h<M

3

11 12

15

10

13 14

9

5 6 7 8

421 3

11 12

14

10

13 15

9

5 6 7 8

421

g�QOh?QOo�R�����X.`C�?QSR[X.R[h � ?t?XZi��.l]t?hGR[qOq¤��n k?R[X�¥IR[XZaUn?iZn?h?oG¸�kG\C�.��k�QSR��Un?h?abVCoNqOQSl]t QS�Zi@MD���/R[XUQS�Zi�kG\C��\pnGl]t��QOXZm_qOQSl.t �.`j±bgI� oNQO�?i1n?h?oNq�\pn?�?qOQSl.t �FQSR[qSR%¾VCoNqOQSl.t?m�R[QOi.R[h<¸Dk�QSR���QOX h?QSl.tji�\pqOqSR�k�n?X.l.t?Y?X.`C�?QSR[X.R[h m�VCh?hGR[h<MÇ�a oj\ph?���ZQSl]tGR[Xb��n �.R[QOh �?X]\pnGl]tji{a�\ph¯R[QOhGR[h � $�� $�� )�¸ \pa¹� R��Zi.R[h¯R[QOhGR[h<¸¦kG\C�b�.`sm_q�\pXbQS�Zi���QSR�kG\C���XZoNn?abR[hjiWa�QOi�k?R[h�P�`Ca�QOhG`N�.i.R[QOhGR[h¾\pn���k?R[a \pa�Y?n?iZQSR[XZi.R[hº¼_l]t&\Cl.tF�?X.R[iZi@M

� R[�/R[hU��QOXD� R[Qj�J` ��k?�)¼_l]t?QSR[� R[Y?n?����qSR k?R[XJqSR�R[X.R[hb¼Fi.R[qOqSR¦k�QSR��Wn?a�abR[X16¸��.`!� R[��QOXZm_iDR[QOhGR¦¥IR[X.�.l.t?QSR[�?n?h?o

R[QOhºo�R[��QS�.�.R��+R[XZaUn?i]\piZQS`Ch k?R[X�r/\pt?qSR[h1�?QS�

16M<P�QSR���h?��\pt?q¦k?R[Xc�DR[XZa�n?i]\piZQS`ChGR[h ��`Ch

ngIqSR[abR[hji.R[h

QS�Zin!M¤� ?X�oNX.`"��R��

nQS�Ziyk�QSR��yR[QOhGR¾�]R[t?X�oNX.`"��R r/\pt?q M g�QOhGRsoNn?i.Rº��Y?Y?X.`�^�QOa�\piZQS`Ch ��QOX.k k�n?X.l.tuk�QSR

��[ R N b R � � � �����&`CXZabR[qJo�R[o�R[�/R[h�Ln! ∼ (

n

e)n√

2πn

¼_`�oNQO�?i!R����/R[Q+¼�\pa �<`��_k?� ¼_l.t?QSR[�/R[Y?n?����qSR�\�Y?XZQS`CXZQ

16! ≈ 2 · 1013

abVCoNqOQSl.tGR��WhG`CX.k�hFn?h?o�R[h<M

6y8�E+7 4 7 0 E¯¶ � � à ¼_R[Qσ ∈ Sn

R[QOhGRy�+R[XZaUn?i]\piZQS`Ch<M g�QOh �+\N\pXi < j

tGR[Q�Ni � $ �'� )�� �"�����`Chσ¸)�¤R[h?h

σ(i) > σ(j)M&P1\C� � !�� � $���� � $�{`_k?R[X � � �(%'�&% k?R[X!�+R[XZaUn?i]\piZQS`Ch

σQS�Zi!k?R�¡Gh?QSR[XZi \pqS�

¼FQOoNh(σ) = (−1)f ∈ {−1,+1},

�¤`C�/R[Qfk�QSRs��h?��\pt?q k?R[X��&R[t?qS�Zi�phGk?Rs��`Ch

σQS�Zi@M �+R[XZaUn?i]\piZQS`ChGR[h«a�QOi�¼FQOoNh

= +1tGR[Q��R[h �'$ ����� $

�+R[XZaUn?i]\piZQS`ChGR[h<¸/k�QSR1a�QOi�¼FQOoNh= −1

�� �'$ ����� $[MJ '�VJ�� &;:=K � P M ��P ∗ �!����� P † �9S����J � � J���:=S �+@�<?I�KM<MY#OVP ∗ �9R�S�" P † �!� �ZY

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¶ �O³ 6y=?> �102 µ�8_79ApBD8FE 8F7�ED8 2 � 8 2p�sÁJ4 = 4 7 0 E à �

��X]\pY?t?QS�.l]t�mN\ph?hya�\ph�R[QOhGRc�+R[XZaUn?i]\piZQS`Chº\pnGl.t°a�QOi �WQOq� R1R[QOhGR�� � ���'$�������������(%'%1�!kG\pX.�Zi.R[qOqSR[h�L

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8��R�k?R[X��GR[t?qS�Zi]\phGk�qOQSR � R[XZiWkG\p�/R[QDR[QOhGR� ��/R[XZmFX.R[n?��n?h?oGMP�n?X.l]t°kG\C� r2�pt?qSR[h�k?R[XB �� R[XZm_X.R[n?��n?h?o�R[h�QS�.i!kG\C�¥I`CXZ�@R[QSl.tGR[h¾R[QOhGR[XW�+R[XZaUn?i]\piZQS`Chº�.l.t?hGR[qOqJ��n�� R��.iZQOa�abR[h<M

6y8�E+7 4 7 0 E¯¶ � � � g�QOhGR � ���� )�5 !() ������!"y`_k?R[Xc\pnGl.t � �'�"� � $���� � ��) ������ ��QS�Zi1R[QOhGR��+R[XZaUn?i]\piZQS`Ch<¸J�¤R[qSl.tGRo�R[h&\pn���� R[QDg�qSR[abR[hFi.RU��R[XZi]\pnG�]l.tji�n?hGk�k?R[h°�WR��Zi � R��Zi!q �C�.�.i@MGP�QSRc·<X]\phG�ZY/`N�ZQOiZQS`Ch<¸&�¤R[qSl.tGR�o�R[h&\pn

in?hGk

j��R[XZi]\pnG�.l.tFi@¸&t&\pi�k�QSR�r��_m�R[qOhG`Ci]\piZQS`Ch

(i, j)M g�QOhGR�·<X]\phG�ZY/`N�ZQOiZQS`Chsk?R[XW�G`CXZa

(i, i + 1)tGR[Q�Ni � � � �H!��"�

�� $�� � �(�*) ���*� ��M

= 4 µ°¶ � � �� ( � & < & : k � ���N43&�> (H" + � : � < ;4'+ � 5 k − 1

� : 5," � � "�(?+ ( � 5'&�5 @� (*( � �B( & � : 5," � � "W(?+ ( � 5 (i, j)(H" + � : � < ;4'+ � 5 2|i− j| − 1 � �L$�1� : #& : +2 " �L$ ;53/N&�5 @

DE&GF�&�(H" @ �9Q � gI��QS�Zi(n1, n2, . . . , nk) = (n1, n2)(n2, n3) . . . (nk−1, nk)

�9QOQ � © R[abR[XZm�RN¸GkG\C�.�(kl)(ik)(kl) = (il)

¼_R[QchFn?h `N© k?�j > i

M ® QOX�� R[� R[QS�.R[h �9QOQ ��a�QOi�£ hGk�n?m_iZQS`Chuh&\Cl.t |i − j| M!£ �Zi |i − j| = 1¸!�.` QS�Zi

(i, j) = (i, i + 1)¸�\pqS�.`�QS�.i�k�QSR�© R[t&\pn?Y?iZn?h?oºm�`CXZX.R[m_i@M+�Wh?o�R[hG`Ca�abR[h<¸�k�QSR�© R[t&\pn?Y?iZn?h?o��ZiZQOa�aUi � ?X

(i, j)M&P1\ph?h

(i, j + 1) = (j, j + 1)(i, j)(j, j + 1).

��qS�.`��ZiZQOa�aUi�k�QSRc© R[t&\pn?Y?iZn?h?o�\pnGl.t#� ?X(i, j + 1)

M ½�gIP

z{0&2p0 ;�;9= 2 ¶ �O³ � & < & ��& :�� N+2 + ( � 5 (H" + � : � < ;4'+ � 5 � : 5," � � "W(?+ ( � 5'&�5 6 " � / : � 5 � �L$�1� : #& : +2 �" �L$ ;53/N&�5 @

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�$� ¶ � �I� " �����U� 6 ��� À�� � � �c� �����y� �������

! 8F79> } 798F;�¶ �O³<5 P�QSRWP1\pX.�Zi.R[qOqOn?h?o�R[QOhGR[X �+R[XZaUn?i]\piZQS`Chya�QOi �WQOq� RWR[QOhGR�� �&\Ck?R[hGk�Q�\poNX]\pa�ab��qOQSR � R[XZi¤k�QOX.R[m_iR[QOhGRcP1\pX.�Zi.R[qOqOn?h?o{\pqS�W��X.`Fk�n?m_i!��`Ch��)\N\pXZ��R[XZi]\pnG�.l]tjn?h?o�R[h<M

1 2 3

321

® QOX�qSR��.R[h°\p��L(132) = (2, 3) ◦ (1, 2)

= 4 µ°¶ �O³ � ��&�(σ ∈ Sn

;5 < σ′ = σ ◦ (a, a+ 1) @ �! 5N5 /�(*>?+

¼FQOoNh(σ) = − ¼FQOoNh (σ′).

DE&GF�&�(H" @ ®¯QOX���R[XZoNqSR[QSl]tGR[h¾k�QSRc�&R[t?qS�Zi�phGk?R1��`Ch σ n?hGk σ′ M�® QOX�n?hji.R[X.�.l]tGR[QSk?R[h¾k�X.R[QD�?�pqOqSRNM��Mi, j

Y&\N\pXZ�¤R[QS�]R���R[X.�]l.t?QSR�k?R[hs��`Cha, a + 1

M P \ph?h¾oNQOqOi�LQS�Zii, j

R[QOh¾�GR[t?qS�Zi]\phGk � ?Xσ¸kG\ph?hº\pnGl.t � ?X

σ′M

© Mi, j

n?hGka, a+ 1

�/R��ZQOiZ�@R[h�R[QOh�o�R[abR[QOhG�²\pabR��Wg�qSR[abR[hFi@¸G��M © MaMGP1\ph?hyoNQOqOi�L

i < aQS�Zi!�GR[t?qS�.i]\phGk � ?X

σo�R[h&\pn�kG\ph?h<¸G� R[h?h

i < a+ 1�GR[t?qS�Zi]\phGk#� ?X

σ′M?�Wh&\pqS`Co � ?X

a+ 1 < iM

��M<£ �Zi {i, j} = {a, a + 1} ¸D�.`�QS�Zi a < a + 1�GR[t?qS�.i]\phGk � ?X

σo�R[h&\pn kG\ph?h<¸Ä� R[h?h

a < a + 1:8F7�E

�GR[t?qS�Zi]\phGk#� ?Xσ′M

® QOXI�.R[tGR[hbt?QSR[X]\pnG��Lσn?hGk

σ′n?hFi.R[X.�.l.tGR[QSk?R[h��ZQSl.t�n?a R[QOhGR[h��GR[t?qS�Zi]\phGkÄMFP�R[X¤¼�\piZ�WQS�ZiIkG\pa�QOiI�/R[��QSR��.R[h<M

½�gIP

z{0&2p0 ;�;9= 2 ¶ �O³ � � " +σ� : � < ;4'+ � 5 n � � $'1� : #& : +2 " � $ ;53/N&�5 � < & : n � : 5," � � "�(?+ ( � 5'&�5 6 " � (H" +

¼FQOoNh(σ) = (−1)n.

��(*5'& ��& :�� N+2 + ( � 5 (H" +�/N& : < & � < & : ;53/N& : < & 6 � & 5 � $ < & � �1!"�( & "W(��L$ >H" � : � < ;4'+ � 5 &�(*5'&�/N& : < &�5

� < & : ;53/N& : < &�5 �E5 � $N> � 5 �� : #& : +2 " � $ ;53/N&�5 < : " +L&�>*> &�5 >�"�" + @

DE&GF�&�(H" @ °QOi�£ hGk�n?mFiZQS`Ch¾h&\Cl.t n Ml��R�k?R � \Cl.tF�&\pXZ��R[XZi]\pnG�.l.tFn?h?o�qOQSR � R[XZi1R[QOhGR[h¾�&\pm_i.`CX −1MP1\C�c¼FQOoNhjn?a

k?R���hGR[n?iZX]\pqSR1g�qSR[abR[hFiWQS�Zi+1M�P1\ �ZR�k?R1·<X]\phG�ZY/`N�ZQOiZQS`Ch���X.`_k�n?m_i!R[QOhGR[X�n?h?o�R[X]\Ck?R[h��Wh?��\pt?qÄ��`Ch ��\Cl]t��

�&\pXZ��R[XZi]\pnG�.l]tjn?h?o�R[hsQS�Zi@¸�oNQOqOiWk�QSR1��nG�.�]\po�Rc\pnGl]t#� ?X!·<X]\phG�ZY/`N�ZQOiZQS`ChGR[h<M ½�gIP

z{0&2p0 ;�;9= 2 ¶ �O³³ � 8;: σ, τ ∈ Sn /�(*>?+�-

¼FQOoNh(σ ◦ τ) =

¼FQOoNh(σ)

¼FQOoNh(τ)

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¶ ��¶ � ; 4 8 2 ED798 2 8_E ~ 8 �<Á ED: 4 7 0 EJ8_E Á E ~ =G; 4 8 2 ED798 2 8FE ~ 8 À�2CÁD}D} 8 �<5

�! #"%$'&�(*),+ 6 < ( & � 1N1#(*> < ;53/ ¼FQOoNh: Sn −→ {−1,+1} = µ2

(H" + &�(*5����������� :�� $N(H" � " � 5 �0: � � &�5 @

DE&GF�&�(H" @ ¼_l.t?X.R[QO�/R σ n?hGk τ \pqS� ��X.`_k�n?mFi ��`Ch�·<X]\phG�ZY/`N�ZQOiZQS`ChGR[hyn?hGk{�¤R[hGk?R k�QSRW��`CXZtGR[XZQOo�R ��nG�.�]\po�R�\ph<M½�gIP

! 8 3 8F79> � ~ =?>�> �I0�� ~�� > ACBD798jH¦8 })Á µ�µF;Â8 Á E+; | >@H+= 2 79> 4P�QSR�abVCoNqOQSl.tGR[h�¥IR[X.�.l]t?QSR[�?n?h?o�R[h��.QOhGkb�+\N\pXZ��R[XZi]\pnG�]l.tjn?h?o�R[h

(i, 16)�9�¤`C�/R[Qk�QSR�abVCoNqOQSl]tGR[h

i I �I��`Ch{k?R[a�ZR[�¤R[QOqOQOo�R[hsrnG�.i]\phGk�\p�?t��ph?o�R[h���M&P1\C�!�<`_l.t

16� \phGk?R[XZi�kG\p�/R[Q ?�/R[X!kG\C�!�GR[qSkÄM

£ �Zi�\pa �Wh�� \ph?o�n?hGk¾\pa ¼_l.t?qOnG�.� kG\C� �<`_l.tsX.R�l.tFi.� n?hji.R[h<¸/�.`�t&\p� R[h¾��QOX�R[QOhGR �'$ ����� $/. � �&�'�G�.`CqSl]tGR[X�)\N\pXZ��R[XZi]\pnG�.l.tFn?h?o�R[h k�n?X.l.t?o�R � ?t?XZi@¸DkG\ �ZR�k?R�¥IR[X.�.l]t?QSR[�?n?h?o°h&\Cl]t `C�/R[h�`Fk?R[Xch&\Cl]t qOQOh?m��ck�n?X.l.t R[QOhGR¥IR[X.�.l]t?QSR[�?n?h?o�h&\Cl]tbn?hji.R[h{� R[��QSR[tFn?h?o��Z�¤R[QS�.R�h&\Cl.t{X.R�l.tFi.�Im�`Ca�Y/R[hG�ZQSR[XZi¦� R[X.k?R[h�aUnG�]�@M�P1\C��¥I`CXZ�@R[QSl.tGR[hk?R[X!� R��]\pa�aUiZY/R[XZaUn?i]\piZQS`Ch�QS�Zi �]`Ca�QOi

+1MFP�QSR�¥IR[XZi]\pnG�.l]tjn?h?o

(14, 15)t&\pi �ZR�k?`Fl]t�¥I`CXZ�@R[QSl]tGR[h −1

MjgI�QS�ZiW\pqS�.`�h?QSl]tji�abVCoNqOQSl.t<¸/\pnG�Zo�R[tGR[hGky��`Ch�k?R[X!��h�� \ph?o��]�ZQOiZn&\piZQS`Ch�hjn?X�L[��n?hGk�L@����ny��R[XZi]\pnG�.l]tGR[h<M&½�gIP

� ��� � � � ��� � ��� ������ � ������ ������� ��� ���� �'�?� ��� � ������������ ��� ��� ���6y8�E+7 4 7 0 E¯¶ �O³/¶ g�QOhGRc�Gn?h?mFiZQS`Ch

F : X ×X −→ RtGR[Q�Ni �"��� $��� ��$ ��$���N¸G� R[h?h

L��F (i, j) = −F (j, i)

Me �F (i, j) 6= 0

� ?Xi 6= j

Mg�QOh�© R[QS�ZY?QSR[q � ?X!R[QOhGRU\pqOi.R[XZh?QSR[X.R[hGk?R��?n?h?m_iZQS`Ch¾\pn��

X = {1, 2, . . . , n} QS�Zi

F (i, j) = i− j. = 4 µ°¶ �O³K � 8;: &�(*5'&���& :�� N+2 + ( � 5 σ ∈ Sn /�(*>?+

i<j

F (σ(i), σ(j)) =¼FQOoNh

(σ)∏

i<j

F (i, j).

Page 82: 10(24350(,76&'*#8)&'*29'*):6;'=@?$?BA7C$=@?$?ED · P QSR W D J / W^_QS ... [tFn?h?o R[h ... [hGR[h ¥I`CXZqSR Zn?h?o R[h°o Rk XZnGl

� � ¶ � �I� " �����U� 6 ��� À�� � � �c� �����y� �������

DE&GF�&�(H" @ ¼_l.t?X.R[QO�/Rc��n?X!¥IR[X.R[QOh�� \Cl.tFn?h?o i′ := σ(i), j ′ = σ(j)MP R[XW��X.`_k�n?mFiW\pn��Ik?R[X!qOQOh?m�R[h¾¼_R[QOi.R � ?� R[X

�)\N\pX.Ri < j

�¦��QOX.k��@R[XZqSR[oNi QOh��.`CqSl]tGRN¸!� ?X k�QSR�\pnGl]ti′ < j′

n?hGk��.`CqSl.tGRN¸ � ?X k�QSRi′ > j′

M_�<R[iZ��i.R[X.R1�)\N\pX.R�ZQOhGk o�R[h&\pn k�QSR��GR[t?qS�Zi�phGk?R���`Ch

σMJ¥IR[XZi]\pnG�.l]tGR[h ��QOX � ?X�k�QSR��]R

f�GR[t?qS�Zi�phGk?R�k�QSR{£ hGk�QO�@R��

in?hGk

�.`�R[XZt&\pqOi.R[h¾��QOX!QOhG�Zo�R��]\pa�iWn?hFi.R[X���nG�Zhjn?iZ��n?h?o�k?R[XW·D\pi.�]\Cl]tGRN¸kG\C�]�F\pqOi.R[XZh?QSR[X.R[hGk�QS�.i@¸GkG\C�W��X.`_k�n?m_i

?�/R[XW\pqOqSRc�+\N\pX.Ri′ < j′

��`ChF (i′, j′)

¸?a�QOi!R[QOhGR[a R�^�iZX]\�¥I`CXZ�@R[QSl.tGR[h(−1)f

M�G`CXZa�\pq)\pnG�Zo�R��.l.t?XZQSR[�/R[h�L

i<j F (i′, j′) =∏

i<j&i′<j′ F (i′, j′)∏

i<j&i′>j′ F (i′, j′)

=∏

i<j&i′<j′ F (i′, j′)∏

j<i&j′>i′ F (j′, i′)

= (−1)f∏

i<j&i′<j′ F (i′, j′)∏

j<i&j′>i′ F (i′, j′)

= (−1)f∏

i<j&i′<j′ F (i′, j′)∏

i>j&i′<j′ F (i′, j′)

= (−1)f∏

i′<j′ F (i′, j′)

= (−1)f∏

i<j F (i, j)

½�gIP

z{0&2p0 ;�;9= 2 ¶ �O³<Ã � " +F

&�(*5'& 1,&�>*( & 1#( /N&= >?+L& : 5N( & : &�5 < & � ;5N4�+ ( � 5 6 " � /�(*>?+�-

¼FQOoNh(σ) =

i<j

F (σ(i), σ(j))

F (i, j)

z{0&2p0 ;�;9= 2 ¶ �O³ � � " �L$�5 &�(*5

� > (�� : $'& : ( /N&�5 � 1;" �L$N5N(?+W+E1,&GF ( &." &�5 �

¼FQOoNh(τ ◦ σ) =

¼FQOoNh(τ) · ¼FQOoNh (σ)

�E5 < & : "0/N&."Z G/�+�-N�B( & � 1N1#(*> < ;53/ ¼FQOoNh: Sn −→ {−1,+1}

(H" + &�(*5 � XZn?Y?Y R[h?tG`Cab`Cab`CXZY?t?QS�Za�nG� @! 8 3 8F79> � ® QOX��.l]t?X.R[QO� R[h ��n?X�¥IR[X.R[QOh�� \Cl.tFn?h?o ��QSR�k?R[X

i′ := σ(i), j ′ := σ(j)n?hGk

i′′ := τ(i′) =

τ(σ(i)), j ′′ := τ(j′) = τ(σ(j))M�© R[abR[XZm�R��¤R[QOi.R[X@¸)kG\C�.��a�QOi

F\pnGl.t k�QSR��?n?h?mFiZQS`Ch k?R�¡Gh?QSR[XZi�k�n?X.l.t

G(i, j) := F (σ(i), σ(j))\pqOi.R[XZh?QSR[X.R[hGk�QS�.i@M¼FQOoNh

(τ ◦ σ) =∏

i<jF (i′′,j′′)F (i,j)

=∏

i<jF (i′′,j′′)F (i′,j′)

i<jF (i′,j′)F (i,j)

=∏

i<jF (i′′,j′′)F (i′,j′)

¼FQOoNh(σ)

=∏

i<jG(τ(i),τ(j))

G(i,j)

¼FQOoNh(σ)

=¼FQOoNh

(τ)¼FQOoNh

(σ)

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¶ �SK � ;Â= 4@0 ED79>@ACBD8 z{|&2p} 8 2 � �

6y8�E+7 4 7 0 E¯¶ �O³ � P�QSR °R[h?o�R�\pqOqSR[X!o�R[X]\Ck?R[h°�+R[XZaUn?i]\piZQS`ChGR[h

An := {σ ∈ Sn |¼FQOoNh

(σ) = 1}QS�Zi!R[QOhGRcÇ�hji.R[XZoNXZn?Y?Y/Rc��`Ch

Snn?hGkytGR[Q�Ni ��� � $��� � $�� $���'$ C � �+5�5 $[M

£ �Zin ≥ 2

n?hGkτ ∈ Sn

R[QOhGRc·JX]\phG�ZY `N�.QOiZQS`Ch<¸G�.`�QS�Zi

τAn = {τσ | σ ∈ An}k�QSR�¾R[h?o�R1k?R[X!n?h?o�R[X]\Ck?RU�+R[XZaUn?i]\piZQS`ChGR[h<M/gI�!QS�ZiW\pqS�.`

Sn = An ∪ τAn, |An| =1

2n!

� ��� � �����(��� �2�;� ��� ����������

gI�)oNQO�?i5Y?q�\pi.`Ch?QS�.l]tGR!TWVCXZY/R[X@¸�kG\C�)tGR[Q�Ni�X.R[o�R[qOa;�&�NQOo�R!�+`Cq���R�k?R[X@¸jk?R[X.R[hbgIl]m�R[h{\pn�� R[QOhGR[XIT n?o�R[q?qOQSR[o�R[h<¸

k?R[X.R[h�¼_R[QOi.R[h�\pqOqSR!oNqSR[QSl]t�\pnG�.�]R[tGR[h<¸�n?hGkb�/R[Q&k?R[hGR[h�\ph �ZR�k?R[X¦gIl.m�R�oNqSR[QSl.t��_QSR[qSR�¼_R[QOi.R[h���nG�]\pa�abR[h?m�`Ca��abR[h<MG��q�\pi.`Ch�t&\pi �#R�k?R[a ��`Ch�QOt?hGR[h�R[QOh�g�qSR[abR[hFi�n?hGkyR[QOhGR[h���oNoNX.R[oj\piZ��nG�Zi]\phGk°��n?o�R�`CX.k�hGR[i@M

��\pabR ¼ �Fa�� `Cq gIqSR[abR[hji@¸ÄrnG�Zi]\phGk g¦l.m�R[h Tc\phFi.R[h �+q �Cl.tGR[h·JR[iZX]\CR�k?R[X � �&R[nGR[X@¸?tGR[Q� � ¬ ���R�^G\CR�k?R[X � g�X.k?RN¸�� R��Zi ¨ L@e ¬Æ�m_i]\CR�k?R[X � �Än���i@¸?oj\C��� VCXZa�QOo ¬ L@e ¨P `_k?R[mC\CR�k?R[X 6 Ç�h?QO��R[X.�Zn?a�¸��piZtGR[XZQS�.l]t eNx L@wNx L@e£�m�`N�]\CR�k?R[X � ® \C�.�]R[X@¸ �� G�.�ZQOo L@e wNx eNx

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�G³ ¶ � �I� " �����U� 6 ��� À�� � � �c� �����y� �������

®�R[h?h ��QOX�R[QOhGR[h Y?q�\pi.`Ch?QS�.l]tGR[h TWVCXZY/R[X�QOa ��\pn?a k�X.R[tGR[h<¸¤kG\ph?h¯� R[X.k?R[h k�QSR°gIl.m�R[h<¸ Tc\phji.R[h n?hGk�+q �Cl.tGR[h Y R[XZa�n?iZQSR[XZi@MJP�QSR�¾R[h?o�R�\pqOqSR[XcabVCoNqOQSl.tGR[h P�X.R[tjn?h?o�R[h �?QOqSk?R[i �#R[� R[QOqS�UR[QOhGR{� XZn?Y?Y RNMÄgI�UoNQO�?i� `CqOo�R[hGk?Rc£ �]`Cab`CXZY?t?QS�ZabR[h�L

= 4 µ°¶ ��¶��

• D(T) = A4

• D(H) = D(O) = S4

• D(I) = D(D) = A5

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� ¶

� � �!� ��(���'���*D( '�-/��� ��� �

� ��� ������� � ��� � � � ��������n

6y8�E+7 4 7 0 E K ��5 ¼_R[QSR[ha, b ∈ Z

M_® QOXWk?R�¡Gh?QSR[X.R[h

a ≡ b mod n ⇔ (a mod n) = (b mod n)

n?hGkº�]\po�R[h �aQS�Zi m�`Ch?oNXZnGR[hjic��n

bab`_k�n?qS`

n�]M ��QSR[XZ� R[Q��ZQOhGk

(a mod n)n?hGk

(b mod n)k�QSR � $$) � $

��`Chan?hGk

b� R[Q<·DR[QOqOn?h?o{k�n?X.l.t

nM_��qOi.R[XZh&\piZQO��RUP R�¡Gh?QOiZQS`Ch�L

a ≡ b mod n ⇔ n | a− b£ �Zi

a ≡ b mod nh?QSl.tji�k?R[X!�&\pqOq ¸&�]`��.l.t?X.R[QO�/R[h���QOX

a 6≡ b mod nM

! 8F79> } 798F;98°K � � �WnG�Wk?R[a �WqOqOi]\po��.qSR[� R[h��.QOhGk���QOXWR���o�R[�¤`Ct?hFiWa�QOiWT�`Ch?oNXZnGR[h?�@R[h°n?a���n?o�R[tGR[h<M

15 ≡ 3 mod 12

365 ≡ 1 mod 7

−4 ≡ 3 ≡ 10 ≡ 17 ≡ 24 mod 7

12331245667 ≡ 7 mod 10

no�R[X]\Ck?R ⇔ n ≡ 0 mod 2

nn?h?o�R[X]\Ck?R ⇔ n ≡ 1 mod 2

= 4 µ�K � � ��53/ &�5 �

��� < ;> � n (H" + &�(*5'& � �jn?QO�N\pqSR[h?��X.R[q�\piZQS`Ch 7 < & : � &�53/N& < & : / 5 �;&�5 �� $N> &�5 @ �! #"$'&�(*),+�-�&."0/�(*>?+

• a ≡ a mod n

• a ≡ b mod n⇒ b ≡ a mod n

• a ≡ b mod n, b ≡ c mod n⇒ a ≡ c mod n

£ hji.R[X.R��.�]\phFiW��QOX.kyR��@¸G�¤R[h?h���QOX�\ph�� \ph?o�R[h�a�QOiWT�`Ch?oNXZnGR[h?�@R[h°��nyX.R�l]t?hGR[h<M

= 4 µ�K �O³ � " +a ≡ b mod n

;5 < c ≡ d mod n 6 " � /�(*>?+

a+ c ≡ b+ d mod n,

a · c ≡ b · d mod n.

Page 86: 10(24350(,76&'*#8)&'*29'*):6;'=@?$?BA7C$=@?$?ED · P QSR W D J / W^_QS ... [tFn?h?o R[h ... [hGR[h ¥I`CXZqSR Zn?h?o R[h°o Rk XZnGl

�&K K z��b�¾À�� �y� �¾���{�U����� ���sÀ

DE&GF�&�(H" @ a ≡ b mod n⇔ a = b+ k · n n?hGk c ≡ d mod n⇔ c = d+ l · n � ?X�o�R[��QS�.�.Rk, l ∈ Z

M?gI�� `CqOoNi�L• a+ c = b+ d+ (k + l) · n ¸G\pqS�.` a+ c ≡ b+ d mod n

M• a · c = (b+ k · n)(d+ l · n) = b · d+ (d · k + b · l + k · l · n) · n ¸G\pqS�]` a · c ≡ b · d mod n

M ½�gIP°QOi!£�hGk�n?m_iZQS`Ch�h&\Cl.t

m� `CqOoNi�\pnG�!�JR[iZ��i.R[X.R[a \pnGl.t

am ≡ bm mod n� ?XW\pqOqSR

m ∈ NM

! 8F79> } 798F;98°K �S¶

• 11 · 11 = 121 = 12 · 10 + 1\pqS�.`

112 ≡ 1 mod 12M

¼_l]t?hGR[qOqSR[X�L11 ≡ −1 mod 12

¸&\pqS�.`112 ≡ (−1)2 = 1 mod 12

M¼_`{\pnGl.t�L

11198 ≡ 1 mod 12M

• 13 · 15 ≡ 1 · 3 ≡ 3 mod 12

• © R��ZiZQOa�abR (134)4 mod 12M134 ≡ 3 mod 12

¸)\pqS�.`1344 ≡ 34 mod 12

M32 ≡ 9 mod 12

¸92 ≡ (−3)2 ≡ 9 mod 12

¸ \pqS�.`1344 ≡ 9 mod 12

M

• 10 ≡ 1 mod 9¸&\pqS�.`

10k ≡ 1 mod 9MGP�QSR�r/\pt?q

. . . a3a2a1 = . . . a3 · 103 + a2 · 10 + a1 ∈ N

QS�.i¤\pqS�.` ≡ . . . a3 +a2 +a1 mod 9��kG\C�¦tGR[Q�Ni@¸jR[QOhGR�r/\pt?qGQS�.iIo�R[h&\pn{kG\ph?h�k�n?X.l]t�v i.R[QOqO�&\pX@¸j�¤R[h?h

QOt?X.R�½�nGR[X.�Zn?a�abRUk�n?X.l]t°v�i.R[QOqO�&\pX!QS�Zi ��M

• © R[iZX]\Cl.tFi.R�k�QSR � ?h���i.R Y � N8S �\[��\pt?q

F5 = 225

+ 1 = 232 + 1 = 4294967297

^B`�bj� N � R[abR[XZm�R@L&R���QS�Zi 64 = 26 ¸&�.`�kG\&�

641 = 10.26 + 1 = 5 · 27 + 1

���/R[XW\pnGl.t641 = 16 + 625 = 24 + 54

® QOX!X.R�l]t?hGR[h �ZR[iZ��imod 641

M&P�QSR1R[X.�Zi.R���qSR[QSl.tFn?h?o{R[XZoNQO�?i�L

5 · 27 ≡ −1 mod 641

��R[t?abR[h���QOXWk�QSR��!tG`Fl]t4¸G�.`���QOX.k�kG\pX]\pnG��L

54 · 228 ≡ 1 mod 641

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K � � 6y8 2�� 8j> 4 :<;9=?>@>�8FE 2 7�ED´Z/nZ

�/Ã

P�QSR1��� R[QOi.R���qSR[QSl]tjn?h?o�qOQSR ��R[XZi54 ≡ −24 mod 641

T�`CaU�?QOh?QSR[X.R[hs��QOX!k�QSR��.RcT�`Ch?oNXZnGR[h?�@R[h<¸�]`�� R[m�`Ca�abR[h���QOX�L

−24228 ≡ 1 mod 641

`_k?R[X232 + 1 ≡ 0 mod 641

��qS�.`�QS�Zi232 + 1

k�n?X.l.t641

i.R[QOqO�&\pX �P�QSRb�/R[mC\ph?hFi.R�·+\pi.�]\Cl.tGRN¸)kG\C�.�

a2 o�R[X]\Ck?R{� R[��M<n?h?o�R[X]\Ck?R�QS�Zi@¸��#Rbh&\Cl.tGk?R[a¿`C�ak�QSR��UQS�Zi@¸<mC\ph?h a�\ph

\pnGl.t°�.` ��`CXZa�n?qOQSR[X.R[h�La2 ≡ a mod 2

P�QSR��.R[X�¼�\Cl.tF��R[XZt&\pqOi q �C�.�Zi!�ZQSl]ty��R[X]\pqOqOo�R[abR[QOhGR[XZh�L = 4 µ�K �SK � z ;98F7�EJ8 2� = 4 µ � 0 E � 8 2C� = 4 ���&�(

p&�(*5'& � : ( � � $N> ;5 < a ∈ Z @ �! 5N5 /�(*>?+

ap ≡ a mod p.

DE&GF�&�(H" @ ®¯QOX��/R[�¤R[QS�.R[h¾k?R[h°¼�\piZ�UR[X.�Zi � ?X a ∈ N0M?® QOX��/R[hjn?iZ�@R[h°£ hGk�n?m_iZQS`Chyh&\Cl.t

aL

��� � a = 0 : 0p ≡ 0 mod p√

��� � ��h?o�R[hG`Ca�abR[h<¸&R���oNQOqOi ap ≡ a mod p� ?X!R[QOh

aM

�� � P1\ph?h�oNQOqOi!a�QOiWk?R[a ©¤QOhG`Ca ��`Ch#��R[��i.`Ch�L

(a+ 1)p = ap + 1 +

p−1∑

n=1

(

p

k

)

ak ≡ ap + 1 ≡ a+ 1 mod p

P1\pa�QOi1QS�Zi k�QSR�© R[t&\pn?Y?iZn?h?o�\pnGl]tºXZQSl.tFiZQOo � ?Xa + 1

M P1\pa�QOi1QS�Zi k?R[X�¼�\piZ��� R[��QSR��]R[h � ?X1\pqOqSRa ∈ N0

MP1\

(−a)p = (−1)pap ≡ (−1)pa mod p¸�oNQOqOiWk�QSR1�WnG�]�]\po�R � ?X�\pqOqSR

a ∈ ZM ½�gIP

� ��� � ��� ��� � � ����� �;��� ���2���Z/nZ

6y8�E+7 4 7 0 E K �9Ã P�QSR ¾R[h?o�R

[a] := a+ nZ := {a+ n · k | k ∈ Z} ⊂ Z

tGR[Q�Ni!k�QSR � $$) �F# ���()L) $ ��`Chaab`Fk�n?qS`

nM[a]

QS�Zi o�R[h&\pn�k�QSR � �jn?QO�N\pqSR[h?��m_q�\C�.�.R ��`Chan?hFi.R[X!k?R[X!TW`Ch?oNXZn��

R[h?��X.R[q�\piZQS`Ch<M

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�$� K z��b�¾À�� �y� �¾���{�U����� ���sÀ

! 8F79> } 798F;98°K ���n = 2

L¦gI��oNQO�?ibhFn?Xb���¤R[Q��WR��ZiZm_q�\C�.�.R[h<¸0 + 2Z = {. . . ,−6,−4,−2, 0, 2, 4, 6, . . .} n?hGk 1 + 2Z =

{. . . ,−5,−3,−1, 1, 3, 5, 7, . . .} M_P1\C�¦�ZQOhGk�\pqS�.`co�R[X]\Ck?R�k�QSR0°R[h?o�R[h�k?R[X¤o�R[X]\Ck?R[h��?���cMjk?R[X n?h?o�R[X]\Ck?R[hr/\pt?qSR[h<M

0+2Z 1+2Z

−5 −4 −3 −2 −1 0 1 2 3 4 5

n = 3L?gI�!oNQO�?iWo�R[h&\pn�k�X.R[QJ�WR��.iZmFq�\C�.�]R[h�L

0 + 3Z = {. . . ,−9,−6,−3, 0, 3, 6, . . .}1 + 3Z = {. . . ,−8,−5,−2, 1, 4, 7, . . .}2 + 3Z = {. . . ,−7,−4,−1, 2, 5, 8, . . .}

0+3Z 1+3Z 2+3Z

−5 −4 −3 −2 −1 0 1 2 3 4 5

= 4 µ�K � � � � 8F> 4 8 Á E ~ � 8j> 4 :<;Â=?>�>@8FE � ��"%/N&�5 n #& : " � $N( & < &�5'& � &."G+ 4 >� #"�" &�5 ��� < ;> � n 6 5 � � >*(�� $< ( & �=>� #"�" &�5

[r] = r + nZ, r ∈ {0, 1, . . . , n− 1}.

� 8;: >*> & a ∈ Z&� 3(H" + ( & : +�/N&�5 &�(*5 r ∈ {0, 1, . . . , n− 1} 6 " � < #"�" a ≡ r mod n @ �B( &." &." r (H" + /N& : < &

(a mod n) @� � & < & � � &." + /N&�$ � : +�/N&�5 &�(*5'& � &."G+ 4 >� #"�" & -

a 7→ [a]

� � & < & : � &." + 4 >� #"�" &0/N&�$ � : +�/N&�5 &�(*5 � &." +�-

[a] 7→ (a mod n)

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K � � 6y8 2�� 8j> 4 :<;9=?>@>�8FE 2 7�ED´Z/nZ

�$�

gI��oNQO�?i�R[QOhGR�mC\phG`Ch?QS�]l.tGR��WR[Y?X.R��.R[hFi]\phji.R[hj� \pt?qIQOh �ZR�k?R[X�T�`Ch?oNXZnGR[h?��mFq�\C�]�.R[a] = a + nZ

¸?h��pa�qOQSl]t(a

mod n)M

6y8�E+7 4 7 0 E K ��5��

Z/nZ :=¾R[h?o�Rck?R[XW��R��ZiZmFq�\C�]�.R[h�ab`Fk�n?qS`

n = {[0], [1], . . . , [n− 1]}® QOX�m�VCh?hGR[h°k�QSR��.R ¾R[h?o�Rck?R[XW��R��ZiZmFq�\C�]�.R[h�a�QOiWk?R[X�°R[h?o�RUk?R[XW�WR��Zi.R QSk?R[hFiZQ ¡G��QSR[X.R[h�L

{[0], [1], . . . , [n− 1]} ∼= {0, 1, 2, . . . , n− 1}��n��¦k?R[XB¾R[h?o�R�k?R[X��WR��ZiZm_q�\C�.�.R[h°��QOX.k#� `CqOo�R[hGk?R[XZa�\&��R[hºR[QOhGR���k?k�QOiZQS`Ch¾n?hGk°R[QOhGR �n?qOiZQOY?qOQOmN\piZQS`Chºk?R��¡Gh?QSR[XZi@M

[a] + [b] := [a+ b],

[a] · [b] := [a · b]

��n��)k?R[X�¾R[h?o�Rck?R[XW��R��Zi.R1�.R[tGR[h�k�QSRc��k?k�QOiZQS`Ch°n?hGkyk�QSR��n?qOiZQOY?qOQOmN\piZQS`Ch°kG\ph?h°�.`b\pnG��L

a+n b := a+ b mod n,

a ·n b := a · b mod n.

Æ ��i�� R[X.k?R[hº��QOX�k?R[hº£ hGk?R�^n�/R[Q

+nn?hGk ·n ��QSR�k?R[X1� R[oNq�\C�.�.R[h<¸Ä\p� R[X kG\ph?hº�.`CqOqOi.R�a�\phs�.QSl.t¾QOa�abR[XkG\pX ?�/R[X!QOa m_q�\pX.R[h��.R[QOh�� \C�!o�R[abR[QOhjiWQS�.i�LG��k?k�QOiZQS`Ch���`Ch¾r/\pt?qSR[h�`_k?R[XW��k?k�QOiZQS`Ch���`Ch°�WR��Zi.R[h<M

! 8F79> } 798F;98°K �955 ® QOX�X.R�l.t?hGR[h¯��nGR[X.�ZibR[QOh?a�\pq!a�QOiZ/2Z

M�gI��oNQO�?i�hjn?X����¤R[Q��WR��Zi.RN¸Ix¾n?hGk LNM)® \C�Y&\C�.�ZQSR[XZi@¸G�¤R[h?h���QOXW�.QSRc\Ck?k�QSR[X.R[h°`_k?R[X�aUn?qOiZQOY?qOQO��QSR[X.R[h±

0 + 0 ≡ 0 mod 2

0 + 1 ≡ 1 + 0 ≡ 1 mod 2

1 + 1 ≡ 2 ≡ 0 mod 2

0 · 0 ≡ 0 mod 2

0 · 1 ≡ 1 · 0 ≡ 0 mod 2

1 · 1 ≡ 1 mod 2

gI��QS�.iW\pqS�.`[1] + [1] = [0]

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® QOX!�.l.t?X.R[QO�/R[h°k�QSR1g�XZo�R[�?h?QS�]�.R1QOh���� R[QJ·D\p�/R[qOqSR[h<¸&R[QOhGRc��k?k�QOiZQS`ChG�Z�¦n?hGk�R[QOhGR�°n?qOiZQOY?qOQOmC\piZQS`ChG�.i]\ ��R[q L

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0 0 1 2 3 4 5 0 0 0 0 0 0 0

1 1 2 3 4 5 0 1 0 1 2 3 4 5

2 2 3 4 5 0 1 2 0 2 4 0 2 4

3 3 4 5 0 1 2 3 0 3 0 3 0 3

4 4 5 0 1 2 3 4 0 4 2 0 4 2

5 5 0 1 2 3 4 5 0 5 4 3 2 1Ç�hGky�.l.t?qOQSR �NqOQSl]t°hG`_l.t � ?X

Z/7ZL

+7 0 1 2 3 4 5 6 ·7 0 1 2 3 4 5 6

0 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0

1 1 2 3 4 5 6 0 1 0 1 2 3 4 5 6

2 2 3 4 5 6 0 1 2 0 2 4 6 1 3 5

3 3 4 5 6 0 1 2 3 0 3 6 2 5 1 4

4 4 5 6 0 1 2 3 4 0 4 1 5 2 6 3

5 5 6 0 1 2 3 4 5 0 5 3 1 6 4 2

6 6 0 1 2 3 4 5 6 0 6 5 4 3 2 1

£ h°k?R[h �n?qOiZQOY?qOQOmN\piZQS`ChG�Zi]\p�/R[qOqSR[h � ?XZ/2Z

¸Z/3Z

n?hGkZ/7Z

�T�pqOqOi1\pn��#¸ kG\C�.� � \pn ��R[X QOh°k?R[X �Wn?qOqO�@R[QOqSRn?hGky���.Y&\pqOi.R�� �#R�k?R[X�®�R[XZiWo�R[h&\pn¾R[QOh?a�\pqJ��`CXZm�`Ca�aUi@M £ hG�Z� R��]`ChGk?R[X.Rc�Zi.R[tji�QOh �ZR�k?R[X r R[QOqSR��9n?hGks¼FY&\pqOi.R��R[QOhGR�LNMFP1\C� tGR[Q�Ni�L ��R�k?R[X�®�R[XZi�\pn ��R[X ��n?qOq/t&\pi R[QOh�a�n?qOiZQOY?qOQOmC\piZQO��R��!£ hj��R[X.�]R��@M�� ?X

Z/6ZQS�Zi kG\C� h?QSl]tji

k?R[XW�&\pqOq M

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K � � 6y8 2�� 8j> 4 :<;9=?>@>�8FE 2 7�ED´Z/nZ

�Ä5

= 4 µ�K ��5 �Z/nZ

(H" + &�(*5 � (*53/ @DE&GF�&�(H" @ P�QSR� W�/R[XZY?X ��9n?h?obk?R[X��!QOh?o�R[QOo�R[hG�.l]t&\ �9i.R[h ?� R[XZq�\C�.�]R[h���QOX!k?R[a �<R��.R[X@M ½�gIP¥IR[XZo�R��.�.R[h���QOX � ?XWR[QOhGR[h °`CabR[hFiWk�QSR�°n?qOiZQOY?qOQOmC\piZQS`Ch�n?hGk��.l.t&\pnGR[h°n?hG�!k�QSRU\p�/R[qS�.l.tGR�� XZn?Y?Y R

(Z/nZ,+) = {[0], [1], [2] = [1] + [1], . . . , [1] + [n− 1] = [0]}\ph<M

= 4 µ�K ��5 � ��&�(ζ = cos( 2πi

n ) + i sin( 2πin ) @

�B( & � 1N1#(*> < ;53/

ψ : Z/nZ −→ µn; ψ([k]) = ζk

(H" +%F�$N> < &��N5N( & : + ;5 < < &��N5N( & : +%&�(*5'&�5 � "

����� :�� $N(H" � " � 5 (Z/nZ,+) 7 (µn, ·) @

DE&GF�&�(H" @ gI�yoNQOqOi�L ζk = ζk′ ⇔ ζk−k

= 1 ⇔ k − k′ ≡ 0 mod n ⇔ [k] = [k′]MW¼_`Ca�QOi�QS�Zi

ψ�¤`Ct?qSk?R�¡Gh?QSR[XZiUn?hGk°QOh �ZR[m_iZQO� M �WqS��QOh �ZR[mFiZQO��R����?�?QOqSk�n?h?o�����QS�.l.tGR[hº��� R[QIR[hGk�qOQSl.tGR[h ¾R[h?o�R[h¾a�QOi oNqSR[QSl]t�_QSR[qSR[hbg�qSR[abR[hji.R[hbQS�.i

ψR[QOhGR�© Q �ZR[mFiZQS`Ch<MNP1\ \pn ��R[X.k?R[a oNQOqOi

ψ([k+l]) = ζk+l = ζk ·ζ l = ψ([k]) ·ψ([l])¸

QS�ZiψR[QOh°��XZn?Y?Y R[h?QS�]`Cab`CXZY?t?QS�ZaUnG�@M ½�gIP

= 4 µ�K ��5F³ � "G+n = p

&�(*5'&�� : ( � � $N> 6 " � (H" + Z/nZ&�(*5�� � :�� & :G@�� ( : " � $ : &�(21,&�5 (*5 < ( &." & � � >*> � $

Fp := Z/pZ.

DE&GF�&�(H" @ P�QSRUR[hji.�.l]tGR[QSk?R[hGk?R�g�QOo�R[hG�]l.t&\ ��iW��`Ch���XZQOa���\pt?qSR[h¾QS�Zi � `CqOo�R[hGk?R

p | a · b⇒ p | a `Fk?R[Xp | b

`_k?R[XW\pqS� ��R[oj\piZQS`Ch ��`CXZa�n?qOQSR[XZi�Lp - a

n?hGkp - a⇒ p - a · b

P�QSR���mN\ph?h�a�\phya�QOiW�WR��.iZmFq�\C�.�]R[h��.`b\pnG�.k�X Gl.m�R[h�L

[a] 6= [0], [b] 6= [0]⇒ [a] · [b] 6= [0]

gI� � `CqOoNi!hFn?h<¸GkG\C�.� � ?X!R[QOh[a] 6= [0]

k�QSR1���?�?QOqSk�n?h?o

[a]· : Z/nZ −→ Z/nZ, [b] 7→ [a] · [b]� �� $ # ��� � QS�Zi@M<P1\yR[QOhGR�QOh �ZR[m_iZQO��Rb�W�?�?QOqSk�n?h?o°��`Ch R[QOhGR[XUR[hGk�qOQSl]tGR[h ¾R[h?o�R�QOh �.QSl.t �]R[qO�G�ZiU\pn?i.`Ca�\piZQS�.l.tR[QOhGR�© Q �#R[m_iZQS`Ch�QS�Zi@¸Cm�VCh?hGR[hb��QOX � `CqOo�R[XZh<¸NkG\C�.��R��IR[QOh

[b] ∈ Z/nZoNQO�?i�a�QOi

[a] · [b] = 1M[a]� R��ZQOiZ��iI�]`Ca�QOi

R[QOhGR1a�n?qOiZQOY?qOQOmC\piZQO��R�£�hF��R[X.�.RN¸GkÄM t<M?QOh�k�QSR��]R[a �&\pqOqJt&\p�/R[hy��QOXWR���a�QOiWR[QOhGR[a � ���5 $ ����n�iZn?h<M ½�gIP

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� � K z��b�¾À�� �y� �¾���{�U����� ���sÀ

6y8�E+7 4 7 0 E K ��5?¶ ¼_R[QRR[QOh��WQOh?o �9m�`Ca�aUn?i]\piZQO�/¸a�QOi

1��M&gIQOhGRcg�qSR[abR[hji

x 6= 0tGR[Q�Ni � � � ��� $�� �($�����`Ch

R¸��¤R[h?h¾R���R[QOh

y 6= 0oNQO�?i@¸G�.`�kG\C�.�

x · y = 0M

= 4 µ�K ��5_K � "G+n43&�(*5'& � : ( � � $N> 6 " � 1,&."W(?+���+ Z/nZ � ;>*>?+L&�(*> & :G@

DE&GF�&�(H" @ £ �Zi n h?QSl.tFiWY?XZQOa�¸�.`�QS�Zi n = a · b a�QOi a, b < nM&¼_`Ca�QOi!QS�Zi

[a] 6= [0], [b] 6= [0]QOh

Z/nZ¸G\p�/R[X

[a] · [b] = [a · b] = [n] = [0].

½�gIP

= 4 µ�K ��5Gà � = 4 µ � 0 E R b�W��+�'� � � � ( � � " + p &�(*5'&�� : ( � � $N> 6 " � /�(*>?+

(p− 1)! ≡ −1 mod p.

� (*( � � " + p &�(*5'& ;53/N& : < & � : ( � � $N> 6 " � /�(*>?+

(−1)p−1

2 (p− 1

2)! ≡ −1 mod p.

DE&GF�&�(H" @�9Q � Lj� ?X

p = 2QS�Zi¤h?QSl.tFi.�¤��n��/R[�¤R[QS�.R[h<¸�\pqS�.`U�]R[Q

p ≥ 3MF£�h�R[QOhGR[a T�VCXZY R[X � R��.QOiZ��i �ZR�k?R��¤g�qSR[abR[hFi

x 6= 0R[QOhy£�hF��R[X.�.R��ya�QOi

x · y = 1M?g�QOhyg�qSR[abR[hji!QS�.i �]R[QOh�R[QOo�R[hGR��!£ hj��R[X.�]R�� o�R[h&\pn�kG\ph?h<¸�� R[h?h

x2 = 1¸�kÄM t<M

�¤R[h?h(x+ 1)(x− 1) = 0

¸?kÄM t<Mx = 1

`_k?R[Xx = −1

M�P�QSR�� � R�k?R[n?i.R[i!QOhG�.� R��.`ChGk?R[X.RN¸?kG\C�]� QOhZ/pZ

oNQOqOi

[2][3] . . . [p− 2] = [1],

kG\ �ZR�k?R��!g�qSR[abR[hFiWa�QOiW�.R[QOhGR[a £ hj��R[X.�]R[h�aUn?qOiZQOY?qOQO��QSR[XZi���QOX.kÄM¼_`Ca�QOi!QS�Zi

[1][2][3] . . . [p− 2][p− 1] = [1][p − 1] = [−1],

� \C��\pqS�!TW`Ch?oNXZnGR[h?��o�R��.l.t?XZQSR[�/R[h(p− 1)! ≡ −1 mod p

�/R�k?R[n?i.R[i@M�9QOQ � LpP \ p−k

2 ≡p+k2 mod p

¸�n?hji.R[X.�.l]tGR[QSk?R[hb�ZQSl.t�k�QSR¦qSR[iZ��i.R[h p−12

g�qSR[abR[hji.R �ZR[�¤R[QOqS�Dn?a R[QOhU¥I`CXZ�@R[QSl.tGR[h��`Ch�k?R[h�R[X.�Zi.R[h p−1

2

M ½�gIP

z{0&2p0 ;�;9= 2 K ��5�� � " +p ≡ 1 mod 4 6 " � /�(213+0&."=&�(*5 a � (?+ a2 ≡ −1 mod p 6 � 5 � � >*(��L$ a = ( p−1

2 )! �J � P 9;Y <MK6S P$Y*P ∗ � � � � P † � �.S

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K � � 6 2 8F79>@= 4 µ ��0 ~ Á ; 0 n Á E ~ � 7�EDBD8F7 4 8FE 7�E Z/nZ� �

� ��� � �����2�;��� ���� ������n���� � �2����� �?�(� � �2�

Z/nZ

! 8F79> } 798F;98°K �95��• 3x ≡ 1 mod 7⇒ x ≡ 5 mod 7

M• 3x ≡ 0 mod 7⇒ x ≡ 0 mod 7

M• 3x ≡ 0 mod 6⇒ x ≡ 0 mod 2

M• 3x ≡ 5 mod 6

t&\pi!m�R[QOhGRc�JVN�Zn?h?o �• 6x ≡ 3 mod 9⇒ x ≡ 2 mod 3

M

= 4 µ�K � ��� �#6 2 8_79>@= 4 µ � 0 ~ Á ; 0 n � �B( & � > &�(�� $ ;53/

a · x ≡ b mod n

$ + /N&�5 < 5N5 &�(*5'& � �G"�;53/ 6 F�&�5N5

ggT (a, n) | b.� 8;: ��F�&�( � �G"�;53/N&�5 x1

;5 < x2/�(*>?+

x1 ≡ x2 mod n,

F�1,&�(

n := n/ggT (a, n) @ � ( : "Z G/N&�5�- < ( & � �G"�;53/ (H"G+ R[QOhGk?R[n?iZQOo ��� < n @DE&GF�&�(H" @• ¼_R[Q ��n?h��Cl]tG�Zi x R[QOhGR1�<VN�Zn?h?o���`Ch a · x ≡ b mod n

M�gI� oNQOqOi�\pqS�]`a · x = b+ k · n a�QOi R[QOhGR[a k ∈ Z

M��n?hyi.R[QOqOi

ggT (a, n)�.`@� `Ct?q

a · x \pqS�W\pnGl.t k · n n?hGk�k?R��Z�¤R[o�R[h°\pnGl]t b M?��qS�.`�oNQOqOi ggT (a, n) | b M

• ¼_R[Q��ZR[iZ��i�n?a�o�R�k�X.R[tFi ggT (a, n) | b ¸ \pqS�.` ggT (a, n) · d = ba�QOi�R[QOhGR[a

d ∈ ZM���\Cl]tsk?R[a P1\pX.�Zi.R[q �

qOn?h?o��.�]\piZ� � ?X!k?R[h�oNo�· oNQOqOiggT (a, n) = A · a+B · n a�QOi A,B ∈ Z

M?��qS�.`�QOhG�Zo�R��²\paUi�L

(A · a+B · n) · d = b

`_k?R[X�n?a�o�R ��`CXZa�ia ·A · d = b−B · d · n ¸�� \C��h?QSl.tFi.��\phGk?R[X.R��W� R�k?R[n?i.R[i�\pqS�

a ·A · d ≡ b mod n.

��qS�.`�QS�ZiA · d R[QOhGRc�<VN�Zn?h?o{k?R[X���qSR[QSl.tFn?h?o a · x ≡ b mod n

M

• ¼FQOhGk�hFn?h x1n?hGk

x2�/R[QSk?R��<VN�.n?h?o�R[h���`Ch

a · x ≡ b mod n¸j�.`�oNQOqOi

a · (x1−x2) ≡ 0 mod n`_k?R[X

\pnGl.ta · (x1 − x2) = k · n � ?X!R[QOh

k ∈ Z.

Page 94: 10(24350(,76&'*#8)&'*29'*):6;'=@?$?BA7C$=@?$?ED · P QSR W D J / W^_QS ... [tFn?h?o R[h ... [hGR[h ¥I`CXZqSR Zn?h?o R[h°o Rk XZnGl

�G³ K z��b�¾À�� �y� �¾���{�U����� ���sÀ

® QOX�i.R[QOqSR[h�� R[QSk?R�¼_R[QOi.R[h�k�n?X.l.tggT (a, n)

n?hGkyR[XZt&\pqOi.R[h

a · (x1 − x2) = k · n

a�QOia = a/ggT (a, n)

M �Wn?h oNQOqOiggT (a, n) = 1

¸+\pqS�.`n | (x1 − x2)

¸J� \C��h?QSl.tFi.��\phGk?R[X.R���tGR[Q�Ni�\pqS�x1 ≡ x2 mod n

M ½�g¦P

z{0&2p0 ;�;9= 2 K � �Ä5� � �B( & � > &�(��L$ ;53/

a · x ≡ 1 mod n

$ +�/N&�5 < 5N5 &�(*5'& � �G"�;53/ 6 F�&�5N5 ggT (a, n) = 1 @ � � >��L$ &�(*5x$'&�(*),+ £�hF��R[X.�.Rc��n

aab`_k�n?qS`

n;5 <

(H" + &�(*5 < &�N+ ( / ��� < ;> � n @� � � 5,"G1,&." � 5 < & : & 6 F�&�5N5 n = p

&�(*5'& � : ( � � $N> (H" + 6 " � (H" + ggT (a, p) = 1 798;: >*> & a 6≡ 0 mod p 6 ;5 <&."=&� 3(H" + ( & : + < 5N5 (

���& : &�(*5'& � 5 #& : " & ��� < ;> � n @

� � � � $ : &�(21,&�5 F ( : a−1 798;:=< ( & ��� < ;> � n &�(*5 < &�N+ ( /N& � 5 #& : " & � a 6 " � 4 � 5N5'&�5 F ( :!< ( & � > &�(�� $ ;53/

a · x ≡ b mod n

> �G" &�5 6 (*5 < & � F ( : >*(*5N4 " ;5 < : & � $ + " � (?+ a−1 �;>?+ (

�>*( � ( & : &�5�-

x ≡ a−1 · a · x ≡ a−1 · b mod n

! 8F79> } 798F;98°K � � �

• £ hj��R[X.�.R mod 7L

1 · 1 ≡ 1 mod 7¸2 · 4 ≡ 1 mod 7

¸3 · 5 ≡ 1 mod 7

a 1 2 3 4 5 6

a−1 1 4 5 2 3 6

�<VN�Zn?h?o���`Ch3 · x ≡ 5 mod 7

QS�Zix ≡ 3−1 · 5 ≡ 5 · 5 ≡ 4 mod 7.

• �WR�l]t?hGR[h mod 101L�<VN�.R1k�QSR�� qSR[QSl.tjn?h?o

5 · x ≡ 16 mod 101.

Page 95: 10(24350(,76&'*#8)&'*29'*):6;'=@?$?BA7C$=@?$?ED · P QSR W D J / W^_QS ... [tFn?h?o R[h ... [hGR[h ¥I`CXZqSR Zn?h?o R[h°o Rk XZnGl

K � � 6 2 8F79>@= 4 µ ��0 ~ Á ; 0 n Á E ~ � 7�EDBD8F7 4 8FE 7�E Z/nZ�¶

® QOX!�ZnGl.tGR[h°��nGR[X.�Zi5−1 mod 101

M© R[abR[XZm�RN¸&kG\C�.�

100 = 20 · 5 + 1.

P1\C��tGR[Q�Ni20 · 5 ≡ −1 mod 101

n?hGk��.`Ca�QOi

5−1 ≡ −20 mod 101.

P�QSR o�R��ZnGl.tFi.R��JVN�Zn?h?obQS�ZiW\pqS�.`

x ≡ −20 · 16 ≡ −17 ≡ 84 mod 101.

• © R[X.R�l]t?hGR k�QSR £ hj��R[X.�.R ��`Ch 30ab`_k�n?qS`

101MCP1\p��n��?X]\pnGl]tGR[h���QOX+k?R[h � $�� $�� )�¸pkG\C�.�

ggT (30, 101) = 1M

¼_l.t&\pnGR[h°��QOX!n?hG�!kG\p��n�k�QSRU·JR[QOqSR[XZm�R[iZi.R�\pnG�Wk?R[a g�n?m_qOQSk�QS�.l.tGR[h¾�WqOo�`CXZQOiZt?a�nG��\ph�L

101 = 3 · 30 + 11

30 = 2 · 11 + 8

11 = 1 · 8 + 3

8 = 2 · 3 + 2

3 = 1 · 2 + 1

2 = 2 · 1 + 0

P�n?X.l.t�� Gl.m��Zn?�G�ZiZQOiZn?iZQS`Ch¾R[XZt&\pqOi.R[h���QOX

1 = 3− 1 · 2= (11− 1 · 8)− (8− 2 · 3)= 11− 2 · 8 + 2 · (11− 1 · 8)= 3 · 11− 4 · 8= 3 · 11− 4 · (30 − 2 · 11)= 11 · 11− 4 · 30= 11 · (101 − 3 · 30) − 4 · 30= 11 · 101 − 37 · 30.

£ h�k?R[XW·D\pi@¸101 · 11 = 1111

n?hGk37 · 30 = 1110

MgI��QS�ZiW\pqS�]`

(30)−1 ≡ −37 mod 101.

6y8�E+7 4 7 0 E K � � �(Z/nZ)∗ := E(Z/nZ)

QS�.i!k�QSR���XZn?Y?Y/R1k?R[XWg�QOh?tGR[QOi.R[h���`ChZ/nZ

M

= 4 µ�K � �?³(Z/nZ)∗ = {[k] | 1 ≤ k ≤ n, ggT (k, n) = 1}

Page 96: 10(24350(,76&'*#8)&'*29'*):6;'=@?$?BA7C$=@?$?ED · P QSR W D J / W^_QS ... [tFn?h?o R[h ... [hGR[h ¥I`CXZqSR Zn?h?o R[h°o Rk XZnGl

�&K K z��b�¾À�� �y� �¾���{�U����� ���sÀ

DE&GF�&�(H" @ [k] ∈ (Z/nZ)∗ ⇔ ∃[x] ∈ Z/nZ : [k] · [x] = [1]⇔ k · x ≡ 1 mod n⇔ ggT (k, n) = 1½�gIP

6y8�E+7 4 7 0 E K � �¶ � � Á ;98 2 >@ACBJ8φ

� �JÁ ED: 4 7 0 E � �¤0G4 798FE 4 � �JÁ ED: 4 7 0 E �

φ(n) := |(Z/nZ)∗| = |{k ∈ {1, 2, . . . , n} : ggT (k, n) = 1}|

n φ(n) ka�QOi

ggT (k, n) = 1

1 0 ∅2 1 1

3 2 1, 2

4 2 1, 3

5 4 1, 2, 3, 4

6 2 1, 5

7 6 1, 2, 3, 4, 5, 6

8 4 1, 3, 5, 7

9 6 1, 2, 4, 5, 7, 8

10 4 1, 3, 5, 7

= 4 µ�K � �GK � = 4 µ � 0 E ^a`8bj� N ��5&à � K�� ��&�( n ∈ N 6 a ∈ Z �(?+ggT (a, n) = 1 @ �! 5N5 /�(*>?+

aφ(n) ≡ 1 mod n.

DE&GF�&�(H" @ £ �Zi ggT (a, n) = 1¸?�.`�QS�.i

[a] ∈ (Z/nZ)∗M?�WqS�.` � `CqOoNi

[a]φ(n) = [a]|(Z/nZ)∗| = [1].

��qS�!TW`Ch?oNXZnGR[h?��o�R��.l.t?XZQSR[�/R[h�QS�Zi!k�QSR��WR[� R[haφ(n) ≡ 1 mod n

M ½�gIP

= 4 µ�K � � à � � 79´8FED>@ApBD= � 4 8_E ~ 8 2 φ � �<Á EJ: 4 7 0 E �� ( � p ��: ( � ⇒ φ(p) = p− 1 @� (*( � p ��: ( � 6 n ∈ N ⇒ φ(pn) = pn − pn−1 @� (*(*( � ggT (n,m) = 1⇒ φ(n ·m) = φ(n) · φ(m)

DE&GF�&�(H" @ �9Q ��\pqOqSR k = 1, 2, . . . , p− 1t&\p�/R[h

ggT = 1a�QOi

pM

�9QOQ �¤gI� QS�ZiggT (k, pn) 6= 1

o�R[h&\pn�kG\ph?h<¸_� R[h?hkk�n?X.l.t

pi.R[QOqO�&\pX QS�Zi@M_£�a £�hFi.R[XZ�C\pqOq {1, 2, . . . , pn} �.QOhGkk�QSR���o�R[h&\pn�k�QSR

pn−1 r/\pt?qSR[h p · l ¸ l ∈ {1, 2, . . . , pn−1} M?gI� � `CqOoNi φ(pn) = pn − pn−1 M�9QOQOQ �JP�QSR��JmC\ph?h�a�\ph�\pnGl]t�k�n?X.l]t��W�?���pt?qSR[h�� R[� R[QS�.R[h<M�®¯QOXD�¤R[X.k?R[h�\p� R[X)�ZY��pi.R[XDR[QOhGR[h��/R��.�.R[X.R[h�© R[� R[QS�o�R[�/R[h<M ½�gIP

Page 97: 10(24350(,76&'*#8)&'*29'*):6;'=@?$?BA7C$=@?$?ED · P QSR W D J / W^_QS ... [tFn?h?o R[h ... [hGR[h ¥I`CXZqSR Zn?h?o R[h°o Rk XZnGl

K �O³ 6y8 2y� B+7�EJ8F>�79>@ApBD8 � 8j> 4 >�= 4 µ �ÄÃ

! 8F79> } 798F;98°K � �$�L���£ �.i

pY?XZQOa�¸Ä�.`�QS�Zi

ap−1 = 1 mod p¸�¤R[h?h

ggT (a, p) = 1MA�n?qOiZQOY?qOQO��QSR[X.R[h���QOX1k�QSR��.R�T�`Ch?oNXZnGR[h?�

a�QOip¸F�.`���QOX.k��ZQSRWiZXZQO�_Q�\pqSR[XZ�¤R[QS�]R o ?qOiZQOo � ?X \pqOqSR

aLap ≡ a mod p

MjP1\C� QS�Zi \p�/R[X k?R[X¤m_qSR[QOhGR ¼�\piZ� ��`ChY � N8S �\[+Me ��¼FQOhGk

pn?hGk

q��XZQOa���\pt?qSR[h<¸ �.`�QS�Zi

φ(p · q) = (p− 1) · (q − 1)M

w �φ(100) = φ(22 · 52) = φ(22) · φ(52) = 22 − 2(52 − 5) = 40

M��qS�.`�oNQOqOi

a40 ≡ 1 mod 100¸G� R[h?h

ggT (a, 100) = 1M

� � £ �Zin = pα1

1 pα2

2 . . . pαk

k

k�QSRc��XZQOa � \pm_i.`CXZ�@R[XZqSR[oNn?h?o{��`Chn¸?�.`�¡GhGk?R[h���QOX

ϕ(n) = ϕ(pα1

1 ) · ϕ(pα2

2 ) · . . . · ϕ(pαk

k )

= (pα1

1 − pα1−11 ) · (pα2

2 − pα2−12 ) · . . . · (pαk

k − pαk−1k )

= pα1

1 · (1−1

p1) · pα2

2 · (1−1

p2) · . . . · pαk

k · (1−1

pk)

= n ·∏

p|n(1− 1

p).

� ��� � ����� � �2��� �;� �;� ��� ��� � � �D��� �¼_R[QOi{��hj�/R[oNQOh¯k?R[X�r R[QOi���QOX.k kG\C�{�<R[�/R[h¯k?R[X�¾R[hG�.l]tGR[h¯k�n?X.l.t � �_mFqOQS�.l]tGR���t��phG`CabR[hGR¾�/R[tGR[XZX.�.l.tFi@MP�QSR��DR[XZQS`_k?R[hº��`Ch ¼_`Ch?hGR�n?hGk#¾`ChGk¾n?hGksk�QSR��1R[�@R[QOi.R[hº� R�R[QOh �GnG�.�]R[hs�GXZnGl.tjiZ�&\pXZm�R[QOi���`Chº�J\phGksn?hGk· QSR[X@MI¼_`Ca�QOi�QS�Zi�R[QOh ¥IR[X.�Zi�phGk�h?QS� ?�/R[X�kG\C��rnG�²\pa�abR[hG�ZY?QSR[q�abR[t?X.R[X.R[X�Y R[XZQS`_k�QS�.l.tGR[X���X.`C�@R��]�.R�R[QOhGR��k?R[XB�pqOi.R��Zi.R[h�a�\piZtGR[a�\piZQS�.l.tGR[hs��X.`C�?qSR[abRNM! 8F79> } 798F; K � ��� P�QSRB¾`ChGk�Y?t&\C�.R[h�t&\p� R[hyR[QOhGR �+R[XZQS`Fk?R���`Ch

29·+\po�R[h<M?�Wh?o�R[hG`Ca�abR[h�R�� QS�Zi!¼_`Ch?hji]\po

n?hGk#��R[n?ab`ChGkÄMG£ hy��QSR[�FQSR[qSR[h�·+\po�R[h#� �pqOqOi�¥I`CqOqOab`ChGk°\pn���R[QOhGR[h�P�QSR[hG�Zi]\poF±¼_R[Q

xk�QSR°o�R��ZnGl]tji.Rº��h?��\pt?q���`Ch ·D\po�R[h<M ��n?a�abR[XZQSR[X.R[h ��QOX�k�QSR°®�`_l.tGR[hji]\po�RN¸��]` � R[X.k?R[h ��QOXy\pn��

� `CqOo�R[hGk?RcTW`Ch?oNXZnGR[h?��oNqSR[QSl]tjn?h?o�R[h�o�R � ?t?XZi�L

x ≡ 2 mod 7, x ≡ 15 mod 29

P�QSR�R[X.�.i.R���qSR[QSl]tjn?h?o���QOX.k{o�R[qSVN�Zi¤k�n?X.l]tx = 2+k · 7 � ?X¤R[QOh

k ∈ Z¸jk�QSR����¤R[QOi.R k�n?X.l]t

x = 15+ l · 29 M��qSR[QSl]tG�.R[iZ�@R[h�qOQSR � R[XZi−7k + 29l = −13

·JR[QOqOn?h?o{a�QOiW�WR��.i qOQSR � R[XZi29 = 4 · 7 + 1,

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�$� K z��b�¾À�� �y� �¾���{�U����� ���sÀ

\pqS�.`�QS�Zi−4 · 7 + 1 · 29 = 1.

°n?qOiZQOY?qOQOmC\piZQS`Ch°a�QOi13qOQSR ��R[XZiWR[QOhGRU�<VN�Zn?h?o

l = 13¸k = 52

M?® QOX ¡GhGk?R[hx = 44

M

! 8F79> } 798F; K � � � � � 7�E 8F7�E�� =?ApBD8 2�z = 2�4 8FE 4@2 79A�: ��NeshFn?a�abR[XZQSR[XZi.R¾¼FY?QSR[qOmN\pXZi.R[h �¤R[X.k?R[h¯QOh � �WR[QOtGR[h ��n«L@wsTc\pXZi.R[h \pnG�Zo�R[qSR[oNi@MIgIQOh rnG�.l.t&\pnGR[X{��QOX.ko�R[�/R[i.R[h<¸_�.QSl.tyR[QOhGR1Tc\pXZi.R���n�abR[XZm�R[h�n?hGk���n��]\po�R[h<¸_QOh��¤R[qSl.tGR[X � 5 �"��� $1�ZQSR�qOQSR[oNi@M?g�X m�VCh?hji.R1�.QSl.t���n?a© R[QS�ZY?QSR[qJk�QSR �GL1abR[XZm�R[h<M

1 2 3 4 5 6 7 8 9 10 11 12 13

14 15 16 17 18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35 36 37 38 39

40 42 43 44 45 46 47 48 49 50 51 5241

P1\ph?hb�¤R[X.k?R[h�k�QSR!Tc\pXZi.R[h�\phGk?R[X.�¦�.`CXZiZQSR[XZi¤n?hGkbk?R[X rnG�.l]t&\pnGR[X �.`CqOqk�QSR��Za�\pq&k�QSR % $ � �($�\ph?o�R[� R[h<¸jQOh�k?R[X�.R[QOhGR1Tc\pXZi.RUqOQSR[oNi@M

1

18

35

52

2

3

4

5 9 13 17

14

33 49

40

27

50

51

4836

39

4420

26

19

34

31

32

23

38

47

46

28

2521

22

29

30

7 11 15

6

8

10

12 16 24

37

42

43

4541

°QOiWk�QSR��.R[h��/R[QSk?R[h°��h?oj\p�/R[h�mN\ph?h�k?R[X�� s\poNQSR[X��U�.l.tG`Ch¾k�QSRco�R[�6�pt?qOi.R�Tc\pXZi.R�\ph?o�R[� R[h<M®¯QSR �9n?h?m_iZQS`p�h?QSR[XZiWkG\C�²±® QOXbhGR[h?hGR[h¯k�QSR#��n?a�abR[X�k?R[X{o�R��ZnGl.tFi.R[h¯Tc\pXZi.R

xMI© R[QOa R[X.�Zi.R[h ¼_l]t?XZQOiZi{��n?X.k?Ryh&\Cl]t¯k?R[X�¼FY&\pqOi.R

o�R ��X]\poNi@MIP�QSR�r/\pt?qSR[h \pn�� k?R[h Tc\pXZi.R[h QOh R[QOhGR[X�¼FY&\pqOi.R�t&\p� R[h t?QSR[Xbk?R[h oNqSR[QSl]tGR[h¯��R��Zi�� R[Q!·DR[QOqOn?h?ok�n?X.l.t L@w ���WR��ZicL QOh°k?R[XcLNM¼FY&\pqOi.RN¸G�WR��.iWe�QOh�k?R[XWe�M¼FY&\pqOi.RN¸<M�M�M ��MGP R[X6� s\poNQSR[X�� �¤R[Q��\pqS�]` L

x ≡ 2 mod 13

Page 99: 10(24350(,76&'*#8)&'*29'*):6;'=@?$?BA7C$=@?$?ED · P QSR W D J / W^_QS ... [tFn?h?o R[h ... [hGR[h ¥I`CXZqSR Zn?h?o R[h°o Rk XZnGl

K �O³ 6y8 2y� B+7�EJ8F>�79>@ApBD8 � 8j> 4 >�= 4 µ ���

£ a¢���¤R[QOi.R[h ¼_l]t?XZQOiZi�R[X�� �pt?XZiUR[Xck�n?X.l]t k�QSRb��h?oj\p� R�k?R[X�r R[QOqSRbk?R[h ��R��Zick?R[X�r/\pt?q¤\pn���k?R[XUTc\pXZi.Rb�/R[Q·JR[QOqOn?h?o�k�n?X.l.t°�VL

x ≡ 1 mod 4

£ h�k?R[Xa°R[h?o�R {1, . . . , 52} oNQO�?i�R���hjn?XWR[QOhGR�r/\pt?q ¸Gk�QSR � R[QSk?R�� qSR[QSl.tjn?h?o�R[hsR[X�� ?qOqOi�L

x = 41

P R[X�·<XZQSl.m�QS�Zi h&\pi ?XZqOQSl.ty�_QSR[qÄR[QOhGk�XZnGl.m��Z��`CqOqSR[X@¸?�¤R[h?h�m�R[QOhGRUr/\pt?qSR[h�\pn��+k?R[hyTc\pXZi.R[h�\pn���o�R�k�XZnGl.m_i!�ZQOhGkĸ�.`ChGk?R[XZh��FQSR[qOqSR[QSl]tji+QOXZo�R[hGk��¤R[qSl.tGR!¼ �_aU�/`CqSRNM@® QOX<�¤`CqOqSR[h�\p� R[XJqOQSR[� R[XDn?hji.R[X.�ZnGl]tGR[h<¸�� \pXZn?a �.`CqSl]tGRI·<XZQSl]m_���n?h?mFiZQS`Ch?QSR[X.R[h�n?hGk�kG\p� R[QJk�QSR1�GX]\po�R��Zi.R[qOqOn?h?o{oNqSR[QSl]t���R[X]\pqOqOo�R[abR[QOhGR[XZh�L

= 4 µ�K � � 5 � � BD7�ED8j>�79>@ACBD8 2�� 8F> 4 >@= 4 µ�� � #" � > &�(�� $ ;53/ "�" �," +L& �

(∗)

x ≡ a1 mod m1

x ≡ a2 mod m2

@@@x ≡ an mod mn

�(?+ggT (mi,mj) = 1 7L8;: i 6= j

$ + (*5 < & : � &�53/N& {0, 1, . . . ,m1 ·m2 · . . . ·mn − 1} &�(*5'& &�(*5 < &�N+ ( /N&� �G"�;53/ @��n�� �¤t?QOhGR��ZQS�]l.t��ZQSR[tFiWk?R[X�¼�\piZ�U�]`b\pnG��L

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5���� K z��b�¾À�� �y� �¾���{�U����� ���sÀ

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M

• ¼_R[iZ�@R Mi := Mmi

� ?Xi = 1, . . . , n

M

• gI�WoNQOqOi ggT (Mi,mi) = 1MG��qS�.`��ZQOhGkyk�QSR���qSR[QSl]tjn?h?o�R[h

Mi · bi ≡ 1 mod mi

qSVN�.�&\pX@MG�<VN�.Rck�QSRU� qSR[QSl.tjn?h?o�R[h �

• ¼_R[iZ�@R Ei := Mi · bi� ?X

i = 1, . . . , nM

• gI�WoNQOqOiW\pqS�.` L

Ei ≡

1 mod mi

0 mod mj, j 6= i

• ��qS�.`�QS�Zix := E1 · a1 +E2 · a2 + . . . +En · an

R[QOhGRU�<VN�Zn?h?ob��`Ch(∗) M

�IÁJ2º� 7�E ~ 8 ÁJ4 79´&:&8_7 4 � ¼_R[QSR[h x1n?hGk

x2��� R[Q �JVN�Zn?h?o�R[h ��n

(∗) MDP1\ph?h oNQOqOi x1 ≡ ai mod min?hGk

x2 ≡ ai mod mi� ?X

i = 1, . . . , n¸F\pqS�.`

x1−x2 ≡ 0 mod miMjP1\C�I�/R�k?R[n?i.R[i \p�/R[X¤\pnGl.t

x1−x2 ≡ 0

mod m1 ·m2 · . . . ·mn = M¸?\pqS�.`

x1 ≡ x2 mod MM?��qS�.`�t&\pi

(∗) QOh�k?R[X�¾R[h?o�R {0, 1, . . . ,M − 1}R[QOhGRcR[QOhGk?R[n?iZQOo�R��<VN�Zn?h?oGM♦

! 8F79> } 798F; � ®¯QOX!�ZnGl]tGR[h°R[QOh x ¸?kG\C� ��`CqOo�R[hGk?R���qSR[QSl.tFn?h?o�R[h°qSVN�Zi�L

x ≡ 1 mod 2

x ≡ 2 mod 3

x ≡ 3 mod 5

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K �O³ 6y8 2y� B+7�EJ8F>�79>@ApBD8 � 8j> 4 >�= 4 µ 5��<5

® QOX�t&\p� R[h°\pqS�]` LM = 30

¸M1 = 15

¸M2 = 10

¸M3 = 6

15 · b1 ≡ 1 mod 2 ⇒ b1 ≡ 1 mod 2

10 · b2 ≡ 1 mod 3 ⇒ b2 ≡ 1 mod 3

6 · b3 ≡ 1 mod 5 ⇒ b3 ≡ 1 mod 5

��qS�.`

x = E1 · a1 +E2 · a2 +E3 · a3

= 15 · 1 + 10 · 2 + 6 · 3≡ 53 mod 30.

��qS�.`�QS�Zix = 23

k�QSRUR[QOhGk?R[n?iZQOo�RU�<VN�Zn?h?o���n(∗) QOh {0, 1, . . . , 29} M

¼FQOhGkAn?hGk

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A × B \pnG�1k?R[X °R[h?o�R�k?R[X�)\N\pX.R

(a, b)¸/�¤`C�/R[Q

a ∈ A n?hGkb ∈ B M/£ �Zi1\pn�� A R[QOhGR�¥IR[XZmFh ?Y��9n?h?o ∗A k?R�¡Gh?QSR[XZi@¸/n?hGk�\pn��

BR[QOhGR

¥IR[XZm_h! ?Y���n?h?o ∗B ¸G�]`�m�VCh?hGR[h���QOX�\pn�� � `CqOo�R[hGk?Rch&\ptGR[qOQSR[o�R[hGk?R�®�R[QS�.RcR[QOhGR1¥IR[XZm_h! ?Y���n?h?o ∗ \pn�� A×Bk?R�¡Gh?QSR[X.R[h�L(a1, b1) ∗ (a2, b2) := (a1 ∗A a2, b1 ∗B b2)

P�QSR ¥IR[XZmFh ?Y��9n?h?o�¡GhGk?R[i�\pqS�.`�m�`Ca�Y/`ChGR[hji.R[hF�¤R[QS�.R��Zi]\piZi@M

= 4 µ�K � � � � (*5 < A ;5 < B � : � � &�5 6 " � (H" + A × B �(?+ < & : � 1N&�5 < &��N5N( & : +L&�5 �'& : 4 5 8 ��7 ;53/ F ( & < & :

&�(*5'& � : � � & @ � (*5 < < ( & �� 4�+ � : &�5 A ;5 < B 31N&�>H" � $ 6 " � (H"G+� � $ A×B 31N&�>H" �L$ @DE&GF�&�(H" @ W�?n?h?o��]\pn���oj\p�/R � ½�gIP��h&\pqS`Co�k?R�¡Gh?QSR[XZi!a�\ph�kG\C����X.`Fk�n?m_i���`Ch

r��XZn?Y?Y/R[h<M

G1 ×G2 × . . . Gr¼FQOhGk

(R,+R, ·R)n?hGk

(S,+S , ·S)�WQOh?o�RN¸Ä�.`�� R[m�`Ca�abR[h���QOXU\pn�� k?R[X�°R[h?o�R

R × S ���¤R[QI¥IR[XZm_h! ?Y����n?h?o�R[h

+n?hGk · L

(r1, s1) + (r2, s2) = (r1 +R r2, s1 +S s2)

(r1, s1) · (r2, s2) = (r1 ·R r2, s1 ·S s2)

= 4 µ�K � �&� �B( & � &�53/N&R× S � (?+ < &�5 � 1N&�5 < &��N5N( & : +L&�5 �'& : 4 5 8 ��7 ;53/N&�5 +

;5 < · (H" + &�(*5 � (*53/ @DE&GF�&�(H" @ W�?n?h?o��]\pn���oj\p�/RNM ½�gIP

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1��M?g�QOhGR1���?�?QOqSk�n?h?o

ϕ : R −→ StGR[Q�Ni

� � �&��!(% !(% !"��5 �'� )$%'�*)�¸��¤R[h?hϕ(a +R b) = ϕ(a) +S ϕ(b)

ϕ(a ·R b) = ϕ(a) +S ϕ(b)

ϕ(1R) = 1S

£ �Ziϕ\pn ��R[X.k?R[a �?Q �ZR[m_iZQO� ¸<�]`yhGR[h?hji�a�\ph

ϕR[QOhGR[h � � �"� ) !(% !"��5 �'� )$%'�*)�`_k?R[XUR[QOhGR[h � ) !(% !"��5 �'� )$%'��)

� !" � � �'$� ���m W�/R[XZqSR[o�R[h¾¼FQSRc�.QSl.t<¸GkG\C�.� ϕ−1 \pn?i.`Ca�\piZQS�.l]t°��QSR�k?R[X�R[QOh��!QOh?oNtG`Cab`Cab`CXZY?t?QS�Za�nG��QS�.i@M �

= 4 µ�K � � ¶ � � BD7�ED8j>�79>@ACBD8 2c� 8j> 4 >�= 4 µ � µ 3 8F7 4 8 � 8 2 >�7 0 E � � " +ggT (n,m) = 1 6 " � (H" + < ( & � 1N1#(*> < ;53/

ϕ : Z/nmZ −→ Z/nZ× Z/mZ

/N&W/N& 1,&�5 < : � $

ϕ(a) := ((a mod n), (a mod m))

&�(*5 � "����� :�� $N(H" � " � 5 � (*53/N&�5 @

= 4 µ�K � � K � � BD7�ED8j>�79>@ACBD8 2�� 8F> 4 >@= 4 µ � ~ 2 7 4�4 8 � 8 2 >�7 0 E � � " + ggT (mi,mj) = 1 798;: i 6= j 6 " � (H" +

Z/m1m2 . . . mrZ∼=−→ Z/m1Z× Z/m2Z× . . . × Z/mrZ

&�(*5 � "����� :�� $N(H" � " � 5 � (*53/N&�5 @

DE&GF�&�(H" @ ¥IR[XZQ ¡G��QSR[X.RN¸&kG\C�.� ϕ(a · b) = ϕ(a) · ϕ(b)M ½�gIP

! 8F79> } 798F; K � � ÃZ/100Z ∼= Z/4Z× Z/25Z

Z/30Z = Z/6Z× Z/5Z = Z/2Z× Z/3Z× Z/5Z

= 4 µ�K � � � ��" " &�( &�5R;5 < S � (*53/N& @ �! 5N5 /�(*>?+ 798;:=< ( & � : � � & < & : ��(*5N$'&�(?+L&�5

E(R × S) = E(R)×E(S).

DE&GF�&�(H" @ (a, b) ∈ E(R×S)o�R[h&\pn�kG\ph?h<¸F�¤R[h?hyR��

(c, d) ∈ R×S oNQO�?i a�QOi (a, b) · (c, d) = (1, 1)M?P�QSR��

�/R�k?R[n?i.R[i�\p� R[XWo�R[h&\pn<¸GkG\C�.�a · c = 1

n?hGkb · d = 1

¸?kÄM ta ∈ E(R)

n?hGkb ∈ E(S)

M ½�g¦P

z{0&2p0 ;�;9= 2 K � � � � " +ggT (m,n) = 1 6 " � /�(*>?+

E(Z/mnZ) = E(Z/mZ)×E(Z/nZ)

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K �O³ 6y8 2y� B+7�EJ8F>�79>@ApBD8 � 8j> 4 >�= 4 µ 5�� �

DE&GF�&�(H" @ E(Z/mnZ) = E(Z/mZ× Z/nZ) = E(Z/nZ)×E(Z/mZ)½�gIP

z{0&2p0 ;�;9= 2 K �O³ � � " +ggT (m,n) = 1 6 " � /�(*>?+ 798;:=< ( & � ;> & : " �L$'& φ � � ;5N4'+ ( � 5�-

φ(m · n) = φ(m) · φ(n)

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f · g = (X3 + 5X + 3) · (X2 + 2)

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= X5 + 7X3 + 3X2 + 10X + 6

� ?XWk�QSR��.Rc��R�l.t?hFn?h?o�R[h�QS�ZiWR��!��VCqOqOQOo�R[oj\pq ¸G� \C�XQS�.i@M?gI�!QS�Zi!R[QOh�� \Cl.t¾R[QOh � � % !�!"� ¸GR[QOhGR � �"������!��($�`_k?R[X

� "! $$#&�" � $[M&P R���� R[QOi.R[X.R[h�t&\p�/R[h���QOX!k�QSRc�+`Ci.R[h?�@R[h���`ChXL

X0 = 1, X1 = X, X2 = X ·X, X3 = X ·X ·X, . . .

a�QOiWk?R[X!R[QOh?��QOo�R[h¾�WR�l.tGR[h?X.R[o�R[qXp ·Xq = Xp+q.

��R[iZ��iWk?R�¡Gh?QSR[X.R[h°��QOX�\pqOqSR���hG`_l.t?a�\pq+R[i�� \C� ��`CXZa�\pqSR[X�L

6y8�E+7 4 7 0 E � ��5 ¼_R[QRR[QOh��!QOh?oGM

• g�QOh � !"� � #!(% a�QOiWTW`_R�ªb��QSR[hji.R[h¾QOhRQS�Zi!R[QOh���nG�.k�XZnGl.m���`Ch�k?R[XW�&`CXZa

f = a0 + a1X + a2X2 + . . . + anX

n

a�QOiai ∈ R

M&P \C�nQS�Zi!kG\p� R[QJh?QSl.tji � R��ZiZo�R[qSR[oNi@M&P�QSR

aitGR[Q��R[h°k�QSR�� ! $�� � � $� � $��k?R��W�+`Cq��FhG`Cab�

fM?��qOi.R[XZh&\piZQO��R�¼_l.t?X.R[QO�F�¤R[QS�.R@L

f =n∑

i=1

aiXi

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5 �$� � 6 ��� � ��� ����� " �{���¾À

• P1\C� � � � � 5 !"� � ! % QS�Zi!kG\C�!�+`Cq��FhG`Ca�¸k?R��.�.R[h�TW`_R�ªb��QSR[hji.R[hº\pqOqSR1oNqSR[QSl]t0�ZQOhGkAL

0 = 0 + 0X + 0X2 + . . .

g¦�!QS�Zi ?�?qOQSl]t<¸G·JR[XZabRUa�QOiWTW`_R�ªb��QSR[hji0�¤R[oN��n?q�\C�.�.R[h<M

• P1\C� � )�5 !"� � #!(% QS�Zi!kG\C�!�+`Cq��FhG`Ca

1 = 1 + 0X + 0X2 + . . .

• R[X]QS�Zi�k�QSR ¾R[h?o�R{\pqOqSR[X��D`Cq��_hG`CabR�a�QOi�T�`FR�ªb��QSR[hFi.R[h QOh

RMJ© R[abR[XZm�RN¸)kG\C�.�U��QOX

R\pqS��k�QSR

·DR[QOqOabR[h?o�R ��`ChR[X]

\pn�<\C�.�.R[h<¸F�¤R[qSl]tGRc\pnG� k?R[h{m�`ChG�Zi]\phFi.R[h��D`Cq��_hG`CabR[h ���+`Cq��FhG`CabR1`Ct?hGRX�

r = r + 0X + 0X2 + . . .

�/R��Zi.R[tFi@M

• . ���"������!" � � $ ��$���� !"� � #!(% $[M&¼_R[Q

f = a0 + a1X + . . . anXn ∈ R[X]

n?hGkg = b0 + b1X + . . . bmX

m ∈ R[X].

® QOXW�.R[iZ�@R[hf + g := (a0 + b0) + (a1 + b1)X + (a2 + b2)X

2 + . . .

• 24�������M5 � � #&������!"�� � $�� $�� � !"� � !(% $[M¼_R[Q

f = a0 + a1X + . . . anXn ∈ R[X]

n?hGkg = b0 + b1X + . . . bmX

m ∈ R[X].

® QOXW�.R[iZ�@R[hf · g := c0 + c1X + c2X

2 + . . .+ anbmXn+m,

� `C� R[Qck =

i+j=k

aibj.

��qS�.`�QS�Zic0 = a0b0 , c1 = a1b0 + a0b1 , c2 = a2b0 + a1b1 + a0b2 , . . .

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5 �Ä5

= 4 µ � � � � (?+ < &�5 � 1,&�5 7 &." + /N&�> &W/�+L&�5�� � & : + ( � 5'&�5 +;5 < · (H"G+ R[X]

&�(*5 � (*53/ @ �! #" � ;>*> � � > �35 ���(H" + 5'&�N+ : > &." ��> & � &�5 +=1N&�� @ + 6�< #" ��(*5," � � > �35 ��� 5'&�N+ : > &." ��> & � &�5 +=1,&�� 8 /�>*(�� $ · @DE&GF�&�(H" @ �DqSR[Q�j\pXZ�/R[QOi@M ½�gIP

! 8F79> } 798F;98 � � �

• C[X]`_k?R[XI\pqOqOo�R[abR[QOh

K[X]� ?X�R[QOhGR[hbTWVCXZY/R[X

K�9k?R[h?hbR[QOhbT�VCXZY R[X�QS�.i+QOhG�.� R��.`ChGk?R[X.R�R[QOh��WQOh?o ��M

• R = C[X]M&P \ph?h�QS�Zi

R[Y ] = C[X,Y ]k?R[XW�+`Cq��FhG`Ca�XZQOh?o�QOhy���¤R[QD¥¦\pXZQ�\p�?qSR[h<M

• �+R[X�£ hGk�n?mFiZQS`Ch°R[XZt&\pqOi.R[h���QOXW�.`�k?R[h��+`Cq��_hG`Ca�XZQOh?o�QOh n ¥¦\pXZQ�\p�?qSR[h�L

R[X1, . . . , Xn] = (R[X1, . . . , Xn−1])[Xn]

6y8�E+7 4 7 0 E � �O³ £ �Zif =

aiXi ∈ R[X]

a�QOian 6= 0

n?hGkak = 0

� ?XW\pqOqSRk > n

¸?�]`�tGR[Q�Ni

LT (f) := anXn

k?R[X � $������ $��$% ��`ChfM<P R[X�T�`FR�ªb��QSR[hFi

LC(f) := antGR[Q�Ni � $����F#&! $ � � � $� �¤��`Ch

f¸Än?hGk kG\C�k¾`ChG`Ca

LM(f) := Xn kG\C� � $ � �W% !" ! %�M�g�QOh��+`Cq��FhG`Ca a�QOiD�JR[QOm�`FR�ªb��QSR[hFi+oNqSR[QSl]t1tGR[Q�Ni !"� % ��$�� � Mp¼_l]t?qOQSR �NqOQSl.t

�.R[iZ�@R[h���QOXGrad(f) = n

M?� ?X!kG\C� ��n?qOqOY `Cq��_hG`Ca k?R�¡Gh?QSR[X.R[h���QOX!X.R[QOh#��`CXZa�\pq

Grad(0) = −∞.

® QOX!�.l.t?X.R[QO�/R[h¾\pnGl]tR[X]<n := {f ∈ R[X] : Grad(f) < n}.

= 4 µ � ��¶

• ��" /�(*>?+

Grad(f + g) ≤ max{Grad(f), Grad(g)}

�(?+ � > &�(�� $N$'&�(?+�/N&�5 < 5N5 F�&�5N5

LT (f) + LF (g) 6= 0 @

• � " +R5 ;>*>?+L&�(*> & :W7 : &�( 6 " � /�(*>?+

LT (f · g) = LT (f) · LT (g),

Grad(f · g) = Grad(f) +Grad(g).

DE&GF�&�(H" @ ��\Cl]t?X.R�l.t?hGR[h<M ½�gIP

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5 � � � 6 ��� � ��� ����� " �{���¾À

z{0&2p0 ;�;9= 2 � �SK ��&�(R5 ;>*>?+L&�(*> & :W7 : &�( @ �! 5N5 (H" +

• R[X]5 ;>*>?+L&�(*> & :W7�: &�( @

• E(R[X]) = E(R) @DE&GF�&�(H" @ ¼_R[QSR[h f, g ∈ R[X]

a�QOif · g = 0

M<¼_R[Q�\pn ��R[X.k?R[aLT (f) = anX

n ¸ LT (g) = bmXm M/P1\ph?h

oNQOqOi0 = LT (f · g) = LT (f) · LT (g) = (an · bm)Xn+m,

\pqS�.`an · bm = 0

�9QOa �!QOh?oR��MGP1\

Rhjn?qOqOi.R[QOqSR[X���X.R[Q+QS�.i@¸���`CqOoNi

an = 0`_k?R[X

bm = 0n?hGk��.`Ca�QOi

f = 0`_k?R[Xg = 0

M¼_R[Q �ZR[iZ��i

f ∈ E(R[X])a�QOiW£ hj��R[X.�.R

g ∈ E(R[X])M&g¦��oNQOqOi�\pqS�.`

f · g = 1n?hGk��]`Ca�QOi

0 = Grad(1) = Grad(f · g) = Grad(f) +Grad(g).

P1\C�1o�R[tji�\p� R[X1hFn?X@¸Ä�¤R[h?hGrad(f) = Grad(g) = 0

M<P�QSR��+`Cq��FhG`CabRfn?hGk

g�ZQOhGk�\pqS�.`�m�`ChG�Zi]\phFi.R

�+`Cq��FhG`CabRN¸Gk�QSR ��QOX a�QOi!g�qSR[abR[hFi.R[h¾\pnG�RQSk?R[hjiZQ ¡G��QSR[XZi�t&\p� R[h<M���qS�.`�R[XZt&\pqOi.R[h���QOX

f, g ∈ E(R)M&¼FQOhGk

n?a�o�R�k�X.R[tjif, g ∈ E(R)

¸GkG\ph?h�h&\pi ?XZqOQSl]t¾\pnGl]tf, g ∈ E(R[X])

M ½�gIP

6y8�E+7 4 7 0 E � �9Ã ¼_R[Qf = a0 + a1X + . . .+ anX

n ∈ R[X]M

• . �*)Z� $�� �W�� �1��`Ch f QOh r ∈ R L

f(r) := a0 + a1r + a2r2 + . . . + anr

n ∈ R

• g�QOhGR � ��� � )�� $�� �($ ��`Ch f QS�ZiWR[QOhr ∈ R a�QOi

f(r) = 0.

© R[abR[XZm�RN¸kG\C�]�W� R[h?hR ⊂ S

¸kG\ph?hR[X] ⊂ S[X]

¸�.`{kG\C�.�!��QOX R[QOhf ∈ R[X]

\pnGl.t¾QOh¾R[QOhGR[as ∈ S\pnG�Z� R[XZi.R[h�m�VCh?hGR[h<¸G`_k?R[XWkG\C�.�

fR[QOhGR ��n?qOqS�Zi.R[qOqSR1QOh

St&\p�/R[h�mN\ph?h<M

! 8F79> } 798F;98 � ���

• (X − 1)(X − 2)(X − 3) ∈ Z[X]t&\pi ��n?qOqS�Zi.R[qOqSR[h

1, 2, 3 ∈ ZM

• 2X − 3 ∈ Z[X]t&\pi�m�R[QOhGR �Wn?qOqS�Zi.R[qOqSRcQOh

Z¸?\p� R[XWR[QOhGR �Wn?qOqS�Zi.R[qOqSR

3/2 ∈ Q ⊃ ZM

• X2 − 2 ∈ Q[X]t&\pi m�R[QOhGR ��n?qOqS�Zi.R[qOqSR[h�QOh

Q¸�\p� R[X ��n?qOqS�Zi.R[qOqSR[h −√2,

√2 ∈ Q(

√2) ⊂ R

M�¼_`�t&\pi\pnGl]t

X2 + 1 ∈ R[X]m�R[QOhGR ��n?qOqS�Zi.R[qOqSR[h�QOh

R¸�\p�/R[X!QOh

CM

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� ��5 � 8F7�; Á EJ´ � 7 4b� 8F> 4 � ��0 ; � E 0� ~ 7 �J79>�7 0 E 5 � �

• � \Cl.t��&R[XZa�\pi!oNQOqOiap ≡ a mod p

M�� ?X�\pqOqSRa ∈ Fp

oNQOqOiW\pqS�]`ap − a = 0

¸GkÄM t<M �"� �$a ∈ Fp

�ZQOhGk��n?qOqS�Zi.R[qOqSR[h�k?R��W�D`Cq��_hG`Cab�

Xp −X = 0.

P1\ � ?X!t&\pi!kG\C�W�+`Cq��_hG`CaXp −X + 1

m�R[QOhGRcR[QOh?��QOo�R ��n?qOqS�Zi.R[qOqSR1QOhFp

M

• P1\C�!�+`Cq��FhG`Ca 2 + 2X ∈ (Z/4Z)[X]t&\pi

1n?hGk

3\pqS� ��n?qOqS�Zi.R[qOqSR[h<M

� ��� � � �2� ����� � � � ��� � ��� �'������� ��� � � � �(����� = 4 µ � � � ��&�(

R&�(*5�5 ;>*>?+L&�(*> & :W7�: &�( & : � (*53/�;5 < g ∈ R[X] �

(?+LT (g) ∈ E(R) @ �! 5N5 /�(213+ &." � � & < & �

f ∈ R[X]&�(*5 < &�N+ ( / 1,&." + (

���+L&q, r ∈ R[X] �

(?+

f = qg + r

;5 <Grad(r) < Grad(g).

DE&GF�&�(H" @ P�QSRUg�QOhGk?R[n?iZQOoNm�R[QOiWQS�Zi�m_q�\pX�L�� \p� R[hy��QOXW���¤R[QJP1\pX.�Zi.R[qOqOn?h?o�R[h

f = q1g + r1 = q2g + r2

��QSRc`C�/R[h�\ph?o�R[o�R[� R[h<¸�.` � `CqOoNi(q1 − q2)g = (r2 − r1).

��nG�!k?R[X���X]\Ck�� `CXZabR[q � `CqOoNi!kG\ph?h

Grad(q1 − q2) +Grad(g) = Grad(r2 − r1).P1\

Grad(r2 − r1) < Grad(g)¸GQS�.i!k�QSR���hjn?X!abVCoNqOQSl]t<¸G� R[h?h

q1 − q2 = 0n?hGk

r2 − r1 = 0M

� ?XUk�QSR�g�^_QS�Zi.R[h?���.`CqSl.t R[QOhGR[X�P1\pX.�Zi.R[qOqOn?h?o°� R[hFn?iZ�@R[h ��QOX�£�hGk�n?m_iZQS`Ch�h&\Cl.tm = Grad(f)

MD¼_R[Qn =

Grad(g)M&®¯QOX!�]l.t?X.R[QO�/R[h

LT (g) = u ·Xn, u ∈ E(R).

£ �Zim < n

¸/�.`��.R[iZ�@R[hº��QOXq = 0

n?hGkr = f

MJ¼_R[QIhFn?hm ≥ n

M/® QOXc�.l.t?X.R[QO�/R[hLT (t) = aXm n?hGk

�/R[iZX]\Cl.tji.R[h kG\C���D`Cq��_hG`Caau−1Xm−ng

MJP R[XU�<R[QOiZi.R[XZa¿QS�Ziau−1Xm−nuXn = aXm ¸+\pqS�.`�k?R[X.�.R[qO�/R

��QSRck?R[X���`ChfM?gI� ��`CqOoNi

Grad(f − au−1Xm−ng) < m.

� \Cl.t�£ hGk�n?mFiZQS`ChG�.��`CX]\pnG�.�.R[iZ��n?h?o�m�VCh?hGR[h���QOXW�.l.t?X.R[QO�/R[h

f − au−1Xm−ng = q′g + r

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5 �G³ � 6 ��� � ��� ����� " �{���¾À

n?hGk��.`Ca�QOif = qg + r

a�QOiq = q′ + au−1Xm−n M ½�gIP

·JR[QOqOn?h?o�a�QOi¤��R��Zi�QOaÅ�+`Cq��_hG`Ca�XZQOh?o ��n?h?mFiZQS`Ch?QSR[XZi o�R[h&\pnG�.`U��QSR!�.l.t?XZQ��iZqOQSl.tGR�� P�QO�_QSk�QSR[X.R[h�QOh{k?R[X ¼_l.tjn?qSRNM��a R[QOh�� \Cl]tG�Zi.R[h�qSR[XZhFi!a�\ph�kG\C��¥IR[X�� \pt?X.R[h°k�n?X.l]t�© R[QS�ZY?QSR[qSR@L! 8F79> } 798F;98 � �95��

f = X5 + 2X4 −X2 + 3X¸g = X2 +X − 1

L

(X5 + 2X4 − X2 + 3X) : (X2 +X − 1) = X3 +X2

−(X5 + X4 − X3)

X4 + X3 − X2 + 3X

− (X4 + X3 − X2)

3X ←��qS�.` L

X5 + 2X4 −X2 + 3X = (X3 +X2)(X2 +X − 1) + 3XM

! 8F79> } 798F; � � f = X5 + 3X4 −X2 + 3X¸g = X2 +X − 1

L

(X5 + 3X4 − X2 + 3X) : X2 +X − 1 = X3 + 2X2 −X + 2

−(X5 + X4 − X3)

2X4 + X3 − X2 + 3X

− (2X4 + 2X3 − 2X2)

−X3 + X2 + 3X

− (−X3 − X2 + X)

2X2 + 2X

− (2X2 + 2X −2)

2 ←��qS�.` L

X5 + 3X4 −X2 + 3X = (X3 + 2X2 −X + 2)(X2 +X − 1) + 2M

! 8F79> } 798F; � � f = X2 + 2¸g = X3 + 5X + 3

MP1\ph?h�QS�Zi

X2 + 2 = 0 · (X3 + 5X + 3) +X2 + 2¸&\pqS�.`

q = 0¸r = f

M

! 8F79> } 798F; ³ � f = X4 − 1¸g = X3 +X2 +X + 1

M

(X4 − 1) : X3 +X2 +X + 1 = X − 1

−(X4 + X3 + X2 + X)

− X3 − X2 − X1 − 1

− (− X3 − X2 − X1 − 1)

0 ←

Page 125: 10(24350(,76&'*#8)&'*29'*):6;'=@?$?BA7C$=@?$?ED · P QSR W D J / W^_QS ... [tFn?h?o R[h ... [hGR[h ¥I`CXZqSR Zn?h?o R[h°o Rk XZnGl

� ��5 � 8F7�; Á EJ´ � 7 4b� 8F> 4 � ��0 ; � E 0� ~ 7 �J79>�7 0 E 5 �¶

��qS�.` LX4 − 1 = (X3 +X2 +X + 1)(X − 1)

¸r = 0

M

= 4 µ � ��5&5 ��&�(R5 ;>*>?+L&�(*> & :W7 : &�( 6 f ∈ R[X]

;5 < α ∈ R @ �! 5N5 /�(*>?+

α(H" + � ;>*>H" +L&�>*> & � 5 f ⇔ (X − α) | f.

DE&GF�&�(H" @ ®¯QOX � ?t?X.R[h�·JR[QOqOn?h?o{a�QOiW�WR��.i!k�n?X.l.t�n?hGk��]l.t?X.R[QO�/R[h

f = q(X − α) + r.

P1\Grad(r) < Grad(X −α) = 1

¸_QS�ZirR[QOhGR TW`ChG�.i]\phji.RN¸GkÄM t<M

r ∈ R MFP�n?X.l.t�gIQOhG�.R[iZ�@R[hy��`Ch α �.R[tGR[h��QOX@¸GkG\C�.�r = f(α)

¸\pqS�.`�t&\p�/R[hy��QOXWR[QOhGRcP1\pX.�Zi.R[qOqOn?h?o

f(X) = q · (X − α) + f(α).

® QOX!�.R[tGR[h��.`Ca�QOi�LαQS�Zi!R[QOhGR ��n?qOqS�Zi.R[qOqSR1��`Ch

f ⇔ (X − α) | f. ½�gIP

z{0&2p0 ;�;9= 2 � ��5 � ��&�(R5 ;>*>?+L&�(*> & :W7 : &�( ;5 < f ∈ R[X] 6 f 6= 0

&�(*5 ��> �N5���

��� � : < 5 @ �! 5N5 1N&."�(?+���+

f$ � �L$," +L&�5,"

n � ;>*>H" +L&�>*> &�5 (*5R @

DE&GF�&�(H" @ °QOi�£ hGk�n?mFiZQS`Chsh&\Cl]t n = Grad(f)Mn = 0

L/g�QOhs�D`Cq��_hG`Ca ��`Ca � X]\Ck¾xbQS�Zi�R[QOhGR�TW`ChG�.i]\phji.R6= 0

¸ \pqS�.`�t&\pifm�R[QOhGR ��n?qOqS�Zi.R[qOqSRNM�Wh?o�R[hG`Ca�abR[h<¸k�QSRU��nG�.�]\po�R QS�Zi!�/R[��QSR��.R[h � ?X�\pqOqSRU�D`Cq��_hG`CabRU��`Ca

��X]\Ck< n

M�£ �Ziα ∈ R ��n?qOqS�Zi.R[qOqSR¦��`Ch

f¸p�.`�QS�Zi

f = (X−α)·g MjgI�+QS�Zi Grad(g) = Grad(f)−1 = n−1¸

\pqS�.`�� R��.QOiZ��igh&\Cl.t�£ hGk�n?m_iZQS`ChG�Z��`CX]\pnG�.�.R[iZ��n?h?o ≤ n− 1

��n?qOqS�Zi.R[qOqSR[h�QOhRM?¼_`Ca�QOi � R��.QOiZ��i

ftGVFl]tG�Zi.R[hG�

1 + (n− 1) = n�Wn?qOqS�.i.R[qOqSR[h�QOh

RM ½�gIP

6y8�E+7 4 7 0 E � ��5 �mα(f) := max{k : (X − α)k | f }

mα(f)tGR[Q�Ni � � $�� � � ���'� $����<`Fk?R[X 24� �����65 ��� � ��� ���/k?R[X ��n?qOqS�Zi.R[qOqSR

αM

mα(f) = 0LαQS�.i�m�R[QOhGR �Wn?qOqS�.i.R[qOqSR1��`Ch

fM

mα(f) = 1LαQS�.i!R[QOh�� \Cl]tGR �Wn?qOqS�.i.R[qOqSR1��`Ch

fM

mα(f) = 2LαQS�.i!k?`CY?Y R[qOi.R ��n?qOqS�Zi.R[qOqSR1��`Ch

fM

¼FQOhGkα1, α2, . . . , αr

k�QSR �Wn?qOqS�.i.R[qOqSR[h ��`Chfa�QOi{¥�QSR[q� \Cl.t?tGR[QOi.R[h

m1,m2, . . . ,mk¸I�.` R[XZX.R[QSl.tGR[h ��QOX

k�n?X.l.t°��QSR�k?R[XZtG`CqOi.R��W��nG�Zi.R[QOqSR[h���`Ch��\pmFi.`CX.R[h�R[QOhGRU�&\pmFi.`CXZQS�.QSR[XZn?h?o

f = g(X − α1)m1(X − α2)

m2 . . . (X − αr)mr ,

�¤`C�/R[Qg ∈ R[X]

R[QOh��D`Cq��_hG`Ca !"�'#$ � ��� � )�� $ � �($�bQS�Zi@M

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5 �&K � 6 ��� � ��� ����� " �{���¾À

= 4 µ � ��5F³ � 8;: &�(*5'&�5�� � :�� & : K "W(*5 < 7 � > /N&�5 < & � "�"Z G/N&�5 ��� ;( > &�5�+�-� � � & < &." � � > �N5 ��� f��� � : < ≥ 1

1N&."W(?+���+ ��(*5 < &."G+L&�5," � &�(*5'& � ;>*>H" +L&�>*> & @� � � & < &." � � > �N5 ��� f

(H" + � : � < ;4'+ � 5 ��(*5'&� :W7 4�+ � : &�5�-

f = c∏

(X − αi), c ∈ K.DE&GF�&�(H" @ �#L�� ⇒ � e � L�¼_R[Q

f 6= 0M�P \ph?h�QS�Zi

f = g ·∏(X −αi)¸F� `C� R[Q

gR[QOhy�D`Cq��_hG`Ca `Ct?hGR��Wn?qOqS�.i.R[qOqSR[h

QS�Zi@M � \Cl.t(1)

a�nG�.�gkG\ph?h�R[QOhGRcTW`ChG�.i]\phji.RU�]R[QOh<M

� e � ⇒ �#L�� L?P�QSR��!QS�.i�mFq�\pX@M ½�gIP

! 8 � 8 2 : Á ED´ � ��5?¶

• g�QOhGR[hºT�VCXZY R[X1a�QOi k�QSR��.R[X g�QOo�R[hG�.l.t&\ ��i1hGR[h?hji a�\ph �"� �'$+!����"� ) ��� ��! � $$) ��� ��!;)$) $�M<P R[X � �(�+5 �F) ��� �� $���. � �'$+! �����/R��]\poNi@¸�kG\C�.�¤k?R[X¤TWVCXZY/R[X

Ck?R[XIm�`Ca�Y?qSR�^?R[hyr/\pt?qSR[hy\pqOo�R[�?X]\pQS�.l.ty\p�?o�R��.l]t?qS`N�.�.R[h�QS�.i@M

• P R[X!TWVCXZY/R[X RQS�Zi h?QSl]tji�\pqOo�R[�?X]\pQS�.l]t¾\p�?o�R��.l]t?qS`N�.�.R[h��?kG\C�!�+`Cq��_hG`Ca

X2 + 1� R��.QOiZ��i m�R[QOhGRcX.R�R[qOqSR

��n?qOqS�Zi.R[qOqSR[h<MGP1\R ⊂ C

¸�QS�ZiW\pnGl.tR[X] ⊂ C[X]

MGgI�!oNQOqOi ��`CqOo�R[hGk?R[X

= 4 µ � ��5_K ��&�(f ∈ R[X] 6 α ∈ C

&�(*5'& � ;>*>H" +L&�>*> & � 5 f @ �! 5N5 (H" +� �L$α&�(*5'& � ;>*>H"G+L&�>*> & � 5 f @

DE&GF�&�(H" @ ¼_R[Q f = anXn + an−1X

n−1 + . . .+ a0¸ai ∈ R

n?hGkα ∈ C

Mf(α) = 0⇔ anα

n + . . .+ a0 =

0⇔ anαn + . . . + a0 = 0⇔ anαn + . . .+ a0 = 0⇔ an(α)n + . . .+ a0 = 0⇔ f(α) = 0M ½�gIP

TW`Ca�Y?qSR�^?R ��n?qOqS�Zi.R[qOqSR[hsm�`Ca�abR[h��.`Ca�QOi�QOa�abR[X1QOhs�)\N\pX.R[hα, α

��`CX@¸ n?hGksR���oNQOqOi(X − α)(X − α) =

X2 − (α+ α)X + αα ∈ R[X]M

z{0 8 � µF7Â8_E 4 8FE Á E ~ ��Á ;�;9> 4 8F;�;98FE¼_R[Q

f = (X−α1)(X−α2) . . . (X−αn) = Xn+a1Xn−1+. . .+an ∈ K[X]

MpP�n?X.l]t��WnG�Za�n?qOiZQOY?qOQO��QSR[X.R[h¡GhGk?R[h���QOX

a1 = −(α1 + α2 + . . . + αn),

an = (−1)nα1α2 . . . αn.

��qOqOo�R[abR[QOhGR[X�k?R�¡Gh?QSR[XZi!a�\ph�L

6y8�E+7 4 7 0 E � ��5&ÃSk :=

i1<i2<...<ik

Xi1Xi2 . . . Xik ∈ Z[X1, X2, . . . , Xn]

= 4 µ � ��5�� ��"%/�(*>?+

ai = (−1)iSi(α1, α2, . . . , αn).

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� ��5 � 8F7�; Á EJ´ � 7 4b� 8F> 4 � ��0 ; � E 0� ~ 7 �J79>�7 0 E 5 �/Ã

� H+;98F7 4�Á EJ´ Á E ~ " 8FB 2 � =?ApB+E Á ;9;9> 4 8_;�;Â8_E® QOX!k?R�¡Gh?QSR[X.R[h°R[QOhGRc���?�?QOqSk�n?h?o

D : K[X] −→ K[X]

��QSR�� `CqOoNi�LD(a0 + a1X + . . . + anX

n) = a1 + 2a2X + . . . nanXn−1

`_k?R[X�m�`Ca�Y&\pmFi.R[X \pn��9o�R��.l]t?XZQSR[� R[h�L

D(∑

i

aiXi) =

i

iaiXi−1

£ �ZifR[QOh��+`Cq��_hG`Ca ��`Ca ��X]\Ck ≤ n ¸F�.`�QS�Zi D(f)

R[QOh��+`Cq��_hG`Ca ��`Ca ��X]\Ck ≤ n− 1n?hGk���QOX.k � !"� % �"�$

. !��($����W� �c��`Chfo�R[h&\ph?hji@M&��qOi.R[XZh&\piZQO��R ��`Ci]\piZQS`Ch�L

f ′ := D(f)

= 4 µ � ��5�� ��"%/�(*>?+�-� ( � D(f + g) = D(f) +D(g)� (*( � D(fg) = D(f)g + fD(g)

DE&GF�&�(H" @ P�QSRU�ÄQOhGR@\pXZQOi�pi (i)QS�Zi!` ÄR[hG�ZQSl]tjiZqOQSl.t<MÄrn

(ii)�/R[abR[XZm�RN¸GkG\C�.�

D(Xn+m) = (n+m)Xn+m−1 = nXn−1Xm +mXnXm−1.

£ �Zif =

i aiXi n?hGk g =

j bjXi ¸?kG\ph?h fg =

i,j aibjXi+j ¸G\pqS�]`

D(fg) =∑

i,j

aibjD(X i+j) =∑

aibjiXi−1Xj + aibjjX

iXj−1 = D(f)g + fD(g).

½�gIP

! 8F79> } 798F;98 � � �$�

• f = X3 +X5 ¸ D(f) = 3X2 + 5X4 M

• f = X3 + x6 + x9 ∈ F3[X]MGP1\ph?h

D(f) = 0 �

• f = Xp −X ∈ Fp[X]¸GkG\ph?h

D(f) = −1M

°QOi!k?R[X ��`CXZa�\pqSR[h°�W�?qSR[QOiZn?h?obm�VCh?hGR[h°��QOX ��R��.i.�Zi.R[qOqSR[h<¸G`C��¾R[t?X�� \Cl]t?hjn?qOqS�Zi.R[qOqSR[h¾��`CXZqOQSR[o�R[h�L

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5 �$� � 6 ��� � ��� ����� " �{���¾À

= 4 µ � � �/5

mα(f) ≥ 2⇔ f(α) = D(f)(α) = 0

DE&GF�&�(H" @ ®¯QOX!�.l]t?X.R[QO� R[h f ′stattD(f)M

=⇒ Lmα(f) ≥ 2⇒ f = (x− α)2h⇒ f ′ = ((x− α)2h)′ = 2(x− α)h+ (x− α)2h′ = (x− α)(2h+

(x− α)h′)M&�WqS�.`��.R[tGR[h���QOX@¸GkG\C�.�

f(α) = f ′(α) = 0M

⇐=Lf(α) = 0⇒ f = (x−α)g

M �WQSR[X]\pnG� � `CqOoNif ′ = g+(x−α)g′

¸F\pqS�]`f ′(α) = g(α)

M_£ �Zif ′(α) = 0

¸�.` � `CqOoNi

g = (x− α)h¸\pqS�.`

f = (x− α)2hM ½�gIP

6y8�E+7 4 7 0 E � � � � g�QOh � \Cl.tFiZX]\po���n¾� XZn?Y?Y R[h�L&¼_R[QGR[QOhGRcR[hGk�qOQSl.tGR���XZn?Y?Y RNM?P�QSR�r/\pt?q

e(G) :=a�\�^ {ord(g) | g ∈ G}

tGR[Q�Ni � 5 !"#$� ���`ChGM

! 8 � 8 2 : Á ED´ � � � � £ �Zih ∈ G ¸��.`�QS�Zi < h > ⊂ G ¸G\pqS�.`��ZQSl.tGR[X ord(h) ≤ |G| ¸&\pqS�]`�oNQOqOi�iZXZQO�_Q�\pqSR[XZ�¤R[QS�]R

e(G) ≤ |G|.

£ �Zie(G) = |G| n?hGk�QS�Zi h ∈ G R[QOh�gIqSR[abR[hji!a�QOi a�\�^�QOa�\pqSR[X�Æ�X.k�hjn?h?oG¸G�]`�oNQOqOi

< h >= G¸�kÄM t<M

GQS�Zi

R[QOhGR � � # ��� ) ��� $W��XZn?Y?Y/R a�QOih\pqS�Wg�XZ�@R[n?o�R[X@M

= 4 µ � � �?³ ��&�(G

&�(*5'&!&�5 < >*(��L$'& \p�/R[qS�.l.tGR � : � � & @ �! 5N5 /�(*>?+

∀g ∈ G : ord(g) | e(G).

DE&GF�&�(H" @ ¼_R[Q h ∈ G R[QOh°g�qSR[abR[hji�a�QOiord(h) = e(G) =: n

¸�n?hGk��.R[Qg ∈ G R[QOh��¤R[QOi.R[X.R���g�qSR[abR[hFi@MgI�

�.R[Qord(g) = m

M&®¯QOX �.l]t?X.R[QO� R[hm = m′ ·m′′ a�QOi m′ = ggT (n,m)

M/gI��oNQOqOi1kG\ph?hord(gm

) = m′′ MP1\

ggT (m′′, n) = 1¸jt&\pi kG\C� g�qSR[abR[hji

gm′′

hk�QSR1�X.k�hjn?h?o

m′′ ·n MFP1\U\p�/R[X n k�QSRWa�\�^_QOa�\pqSR�abVCoNqOQSl.tÆ�X.k�n?h?ob� \pX@¸ � `CqOoNim′′ = 1

¸?kÄM t<Mm = ggT (n,m)

M&P�QSR����/R�k?R[n?i.R[i�\p�/R[X@¸GkG\C�.�m | n M ½�gIP

! 8 � 8 2 : Á ED´ � � �¶ P�QSR��.R��WnG�]�]\po�R�oNQOqOi�Q M ��M h?QSl.tFi � ?X h?QSl]tji�\p�/R[qS�.l.tGR�� XZn?Y?Y R[h<MÄ£ hS3

oNQO�?icR���hFn?Xg�qSR[abR[hFi.RUk?R[X��X.k�hFn?h?o

1¸2n?hGk

3M

= 4 µ � � �GK ��&�(K

&�(*5 � � :�� & : ;5 < G ⊂ K∗ = K \ {0} &�(*5'& &�5 < >*(�� $'& ��5 +L& : / : � � & @ �! 5N5 (H" +G

� �N4 >*(H" �L$��

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� ��5 � 8F7�; Á EJ´ � 7 4b� 8F> 4 � ��0 ; � E 0� ~ 7 �J79>�7 0 E 5 �$�

DE&GF�&�(H" @ ¼_R[Q n := e(G)��QSR1`C�/R[h<MGP1\

G\p�/R[qS�.l.t�QS�Zi@¸�oNQOqOi

ord(g) | n � ?X�\pqOqSRg ∈ G M���QSR[X]\pnG� ��`CqOoNi

gn = 1 ∀ g ∈ G.

P1\C�!�+`Cq��FhG`CaXn − 1

t&\pi!QOhKtGV_l.tG�.i.R[hG�

n��n?qOqS�Zi.R[qOqSR[h<MG��qS�.`

|G| ≤ n = e(G).

P1\�iZXZQO�FQ�\pqSR[XZ� R[QS�.R |G| ≥ e(G)¸?a�nG�.����qSR[QSl]t?tGR[QOiWo�R[qOi.R[h<¸?n?hGk��.`Ca�QOi!QS�Zi

G� �_mFqOQS�]l.t<M ½�gIP

! 8F79> } 798F;98 � � �/Ã�9Q �K = C∗ ¸ G = µn

Mµn

QS�.i!QOh�k?R[XW·D\pi�R[QOhGR1� �Fm_qOQS�.l]tGRU��XZn?Y?Y RNM�9QOQ �

K = Fp¸G = F∗

p

M?gI� � `CqOoNi�LF∗p

QS�Zi�� �_m_qOQS�.l.t � P�M t<MGR���oNQO�?i h ∈ F∗p

¸G�]`�kG\C�.�

F∗p = {1, h, h2, . . . , hp−1 = 1}.

= 4 µ � � ��� ��"%/�(213+%&�(*5'&�5 � : � � &�5N(H" ����� :�� $N(H" � "

E(Z/pZ) −→ Z/(p− 1)Z.

6y8�E+7 4 7 0 E � � �$� P1\C� � $ �'$ ��"� $�� � % !�!"� (a

p

) ��na ∈ Z

n?hGk�k?R[X n?h?o�R[X]\Ck?R[h���XZQOa���\pt?qpQS�Zi�k?R�¡Gh?QSR[XZi

k�n?X.l.t(

a

p

)

∈ {1,−1}, ap−1

2 ≡(

a

p

)

mod p.

= 4 µ � � � � �! #" � &W/N&�5 <�: & � � � � 1 � >�(H" + &�(*5 � : � � &�5N$ ��������� :�� $N(H" � "

F∗p −→ {−1,+1},

(

a

p

)

·(

b

p

)

=

(

ab

p

)

.

(

a

p

)

= 1 ⇔ ��"%/�(213+ &�(*5b ∈ Z �

(?+b2 ≡ a mod p.

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5 � � � 6 ��� � ��� ����� " �{���¾À

� EJ=G; 0 ´ 798�µ 3 79>@ACBJ8_EZÁ E ~ K[X] �

�WQOh?oZ K[X]g�qSR[abR[hFi

nr/\pt?q �)�+`Cq��_hG`Ca

g�QOh?tGR[QOi.R[h −1,+1 K∗·DR[QOqO�&\pXZm�R[QOi

(m 6= 0), n | m (g 6= 0), f | g⇒ |n| < |m| ⇒ Grad(f) < Grad(g)·DR[QOqOn?h?o�a�QOiW��R��Zi

¾R[h?o�Rck?R[XW��R��Zi.R {0, 1, . . . , n− 1} K[X]<n��XZQOa���\pt?qp ??��XZQOa � \pm_i.`CXZ�@R[XZqSR[oNn?h?o

T�`Ch?oNXZnGR[h?��X.R�l.t?hjn?h?o13 ≡ 5 mod 8 ??��R��ZiZmFq�\C�]�.R[h?XZQOh?o

Z/nZ ??T�VCXZY R[XZ/pZ ??

® QOX��¤R[X.k?R[h���R[X.�.nGl.tGR[h<¸&k�QSRcX.R�l.tji.R�¼FY&\pqOi.R1��n�R[XZo��ph?�@R[h �

6y8�E+7 4 7 0 E � � � 5 ¼_R[QRR[QOh��!QOh?oG¸

r ∈ R ¸ r 6= 0¸r /∈ E(R)

MrtGR[Q�Ni � ����$ �(� � �X! $�� ¸�¤R[h?h

r = a · b⇒ a ∈ E(R)`_k?R[X

b ∈ E(R).

£ �Zirh?QSl]tjiWQOXZX.R�k�n?��QO�/R[q ¸&�]`�tGR[Q�Ni

f��$B� � � �X! $ � M P1\ph?h�QS�Zi

r\pqS�.`�kG\pX.�Zi.R[qOqO�&\pX�\pqS�

r = a · b,

�¤`C�/R[Q<� R�k?R[XahG`_l.t

bgIQOh?tGR[QOi.R[h�QOh

R�ZQOhGkÄM

! 8F79> } 798F;98 � � � �

• P�QSR1QOXZX.R�k�n?��QO�?qSR[h¾g�qSR[abR[hji.RUQOh Z�ZQOhGk ±p a�QOi p Y?XZQOa�M

• g�QOh��+`Cq��FhG`Ca f ∈ K[X]QS�ZiDQOXZX.R�k�n?��QO� R[q ¸��¤R[h?h�\pnG�

f = g·h � `CqOoNi@¸�kG\C�]�g`Fk?R[X

hR[QOhGR¤TW`ChG�Zi]\phFi.R

QS�.i@M��WhGk?R[X.� \pnG�Zo�R�k�X Gl.m_i�LfQS�Zi�QOXZX.R�k�n?��QO�/R[q ¸�� R[h?h

fh?QSl]tji ��X.`Fk�n?m_iI��`Ch��+`Cq��_hG`CabR[h{��`Ca � X]\Ck

< Grad(f)QS�Zi@M)TW`ChG�Zi]\phFi.R��+`Cq��FhG`CabR

f = c, c ∈ K���pt?qSR[h h?QSl.tFi{\pqS��QOXZX.R�k�n?��QO� R[q ¸I�¤R[QOqW�ZQSR

oNqSR[QSl]t0`_k?R[X!R[QOhGRcg�QOh?tGR[QOi!QOh

K[X]�ZQOhGkÄM

• P1\C�!�+`Cq��FhG`Ca (X − α)QS�Zi�QOXZX.R�k�n?��QO�/R[q M

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� ��5 � 8F7�; Á EJ´ � 7 4b� 8F> 4 � ��0 ; � E 0� ~ 7 �J79>�7 0 E 5 � 5

• g�QOh �D`Cq��_hG`Ca f��`Ca ��X]\Ck ≤ 3

QS�Zi1o�R[h&\pn�kG\ph?hºX.R�k�n?��QO�/R[q ¸Ä� R[h?h R��1��X.`_k�n?mFi1QS�Zi1��`Ch�R[QOhGR[a�+`Cq��_hG`Ca¹��`Ca¹��X]\Ck

1n?hGk R[QOhGR[a¿�¤R[QOi.R[X.R[h �&\pm_i.`CX@M)© R��ZQOiZ��i

f#�$�� $ � ��� � )�� $ � �($ �

K¸+�.`�QS�.i

fQOXZX.R�k�n?��QO�/R[q M¤¼_`��ZQOhGk

X2 + 1 ∈ R[X], X2 − 2 ∈ Q[X], X3 − 2 ∈ Q[X], X2 + X + 1 ∈F2[X], X3 +X + 1 ∈ F2[X], X3 + 1 ∈ F3[X]

QOXZX.R�k�n?��QO�?qSRU�+`Cq��FhG`CabRNM

• P1\C���D`Cq��_hG`Ca f = X4 + 3X2 + 2 ∈ R[X]QS�Zi�X.R�k�n?��QO� R[q ¸

f = (X2 + 1)(X2 + 2)¸\p�/R[X

ft&\pi

` ÄR[hG�ZQSl]tjiZqOQSl.t¾m�R[QOhGR ��n?qOqS�Zi.R[qOqSR1QOhRM

= 4 µ � � �&� & < &." � � > �N5 ��� f ∈ K[X] �(?+Grad(f) ≥ 1

(H"G+ � : � < ;4'+ � 5 ( :W: & < � (21#> &�5 � � > �N5 ��� &�5 @DE&GF�&�(H" @ �QOi�£ hGk�n?mFiZQS`Ch h&\Cl.t

n = Grad(f)Mn = 1

LW\pqOqSRsqOQOhGR@\pX.R[h �D`Cq��_hG`CabR �.QOhGk QOXZX.R�k�n?��QO� R[q M��h?o�R[hG`Ca�abR[h<¸jk�QSR���nG�.�]\po�R QS�Zi�� \pt?X � ?X¦\pqOqSR��+`Cq��FhG`CabRW��`CaÅ��X]\Ck

< nMCg�hji �¤R�k?R[X

fQS�.i�QOXZX.R�k�n?��QO� R[q ¸

`_k?R[Xf = g · h a�QOi Grad(g), Grad(h) < n

M?P1\ph?h��ZQOhGkya�QOiW£ hGk�n?m_iZQS`Ch°\p� R[Xgn?hGk

h��X.`Fk�n?m_i.R1��`Ch

QOXZX.R�k�n?��QO�?qSR[h°�+`Cq��_hG`CabR[h<¸&n?hGky�.`Ca�QOi�\pnGl.tfM ½�gIP

6y8�E+7 4 7 0 E � � � ³ ¼_R[QRR[QOh �WQOh?o�`Ct?hGR���n?qOqOi.R[QOqSR[X@¸<n?hGk �.R[QSR[h

a, b ∈ R ¸<h?QSl.tji��/R[QSk?R�oNqSR[QSl.t x�MJg�QOhg�qSR[abR[hFic ∈ R tGR[Q�Ni ��� �°��`Ch

an?hGk

b¸��¤R[h?h

L��c | a n?hGk c | b Me �c′ | a n?hGk c′ | b⇒ c′ | c

! 8 � 8 2 : Á ED´ � � � ¶ P1\an?hGk

bh?QSl]tjiW�/R[QSk?R

0�ZQOhGkĸ�� `CqOoNiW\pnG� �#L���¸?kG\C�]�

c 6= 0M

g�X�� ?qOqSR[hcn?hGk

d� R[QSk?R�k�QSRU© R�k�QOh?oNn?h?o�R[h �#L�� n?hGk � e ��¸�.` � `CqOoNi

c | d n?hGk d | c MP�M t<M L d = u · c n?hGkd = v · c � ?X

u, v ∈ R M���QSR[X]\pnG� � `CqOoNi c = uvc`_k?R[X

c(1 − uv) = 0MP1\

c 6= 0¸���`CqOoNi!t?QSR[X]\pnG�

1 = uv¸

kÄM t<Mun?hGk

v�ZQOhGkyg�QOh?tGR[QOi.R[h°��`Ch

RM?gIQOh

ggT��� R[QSR[X�g�qSR[abR[hFi.R

an?hGk

bQS�ZiW\pqS�]`?¸��¤R[h?h�R[X!R�^�QS�ZiZQSR[XZi@¸

!�� ) �(�$� $�� $ � � $ � �ÄR[QOhGk?R[n?iZQOoGM£�a �WQOh?oZm�VCh?hGR[h���QOX � R��ZiZqSR[o�R[h�L

ggT > 0® QOX¤� R[X.k?R[h�oNqSR[QSl]t��.R[tGR[h<¸_kG\C�.�¦QOaÈ�WQOh?oK[X]

k?R[XggT

��� R[QSR[X gIqSR[abR[hji.R1R�^�QS�ZiZQSR[XZi@¸_n?hGk���QOX m�VCh?hGR[hk?R[h

ggT\pqS� !"� % ��$�� � � R��ZiZqSR[o�R[h<M

6y8�E+7 4 7 0 E � � � K ¼_R[QSR[hf, g ∈ K[X]

¸?h?QSl]tjiW�/R[QSk?R0M¼_R[iZ�@R

T (f, g) := {h ∈ K[X] : h | f n?hGk h | g}.T q�\pX!QS�Zi

T (f, g) = T (f − q · g, g).£ �Zi!\pqS�.`

g 6= 0n?hGk � ?t?X.R[hy��QOX!·JR[QOqOn?h?o{a�QOi!�WR��Zi!k�n?X.l]t

g\pnG�@¸

f = qg + r¸��.`�QS�.i

T (f, g) = T (r, g)M

P�QSR�� � ?t?XZi@¸?o�R[h&\pn���QSR1QOhZ¸���n?a ��# � �X�"�7) ��� $�4. � � !"��� � �*%3�*) QOh

K[X]L

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5 � � � 6 ��� � ��� ����� " �{���¾À

• a1 := f¸a2 := g

� 6= 0��M

• � ?X i = 2, 3, . . . : ai+1 := (ai−1 mod ai)

• £ �Zi ak+1 = 0¸&�.`�QS�Zi

akR[QOh

ggT��`Ch

fn?hGk

gM

s\ph�¡GhGk?R[i0�pt?h?qOQSl.t���QSR1QOhZL

• Grad(a2) > Grad(a3) > . . .¸

\pqS�]`�aUnG�.�!h&\Cl.t°R[hGk�qOQSl]t��_QSR[qSR�¼_l.t?XZQOiZi.R[hsx�\pn��9iZX.R[i.R[h<M

• T (a1, a2) = T (a2, a3) = . . . = T (ak, 0)¸<�¤`CX]\pnG����QOX��]R[tGR[h<¸<kG\C�.�

ak�.`��¤`Ct?q

a1\pqS��\pnGl.t

a2i.R[QOqOi@¸?�]`Coj\pX!kG\C�.� �#R�k?R[X�·DR[QOqSR[X!��`Cha1n?hGk

a2\pnGl]t

aki.R[QOqOi@M?P1\C� tGR[Q�Ni@¸

akQS�Zi

ggT��`Ch

a1n?hGk

a2M

= 4 µ � � � à � ��2�3 8_7 4 8 2�4 8 2¾��Á :<;�7 ~ 79>@ACBJ8 2�� ;9´ 02 7 4 B �sÁ > �¤´&´ � �]6y= 2 > 4 8F;�; Á ED´&>@>�= 4 µ�� ��&�( &�5f, g ∈

K[X] 6 5N(�� $�+E1,&�( < & 0 @ �! 5N5 /�(213+0&."A,B ∈ K[X] 6 " ��< #"�"

ggT (f, g) = A · f +B · g.

DE&GF�&�(H" @ ¥�VCqOqOQOo{\ph&\pqS`Co���n?a © R[�¤R[QS�!QOhZM ½�gIP

= 4 µ � � � � � �c2 7 � 8F79´&8_EJ>�ApBD= � 4 7 2p2 8 ~ Á µj79H+;98 2 ��0 ; � E 0 � 8 � ��&�(f( :W: & < � (21,&�> @ �! 5N5 /�(*>?+

f | g · h⇒ f | g `_k?R[Xf | h.

DE&GF�&�(H" @ �Wh?o�R[hG`Ca�abR[h<¸ f - gM&P�R[X

ggT k��`Ch

fn?hGk

gq �C�]�Zi!�ZQSl.t°kG\pX.�Zi.R[qOqSR[h°\pqS�

k = A · f +B · g.

P1\fQOXZX.R�k�n?��QO�/R[q&QS�Zi@¸Na�nG�.�

kR[QOhGR!g�QOh?tGR[QOi �.R[QOh<M�·DR[QOqOn?h?o�k�n?X.l]t

kR[XZoNQO�?i R[QOhGRW�WR[q�\piZQS`Ch

1 = Af + BgM

°n?qOiZQOY?qOQOmC\piZQS`Ch°a�QOihqOQSR � R[XZi

h = Afh+ Bgh.

��QSR[X]\pnG� ��`CqOoNif | h M ½�g¦P

z{0&2p0 ;�;9= 2 � � � � ��&�(f( :W: & < � (21N&�> @

f | g1g2 . . . gr ⇒ ∃i : f | gi

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� ��5 � 8F7�; Á EJ´ � 7 4b� 8F> 4 � ��0 ; � E 0� ~ 7 �J79>�7 0 E 5 ��

= 4 µ � �O³�� & < &." � � > �N5 ��� f ∈ K[X](H" + < : " +L&�>*>21� : >H"

f = cg1g2 . . . gr

�(?+c ∈ K∗ ;5 < gi ( :W: & < � (21,&�> ;5 < 5

� :�� ( & : + � < #"E$'&�(*),+ � (?+ � &�(?+ 4 � &���� ( &�5 + � � @ �B( &." &B� : "G+L&�>*> ;53/(H" + 1#(H"E 7�< ( & �!&�(*$'&�5 7 � > /N& < & : gi �

"=&�(*5 < &�N+ ( / @DE&GF�&�(H" @ ®¯QOX1t&\p�/R[h �.l]tG`Ch o�R��.R[tGR[h<¸JkG\C�.� �ZR�k?R�� f �.`�R[QOhGRbP1\pX.�Zi.R[qOqOn?h?o°� R��ZQOiZ��i@MJ�Wh?o�R[hG`Ca�abR[h<¸J��QOXt&\p�/R[h����¤R[QJ�]`CqSl.tGR[XWP1\pX.�Zi.R[qOqOn?h?o�R[h�L

cg1g2 . . . gr = c′h1h2 . . . hs

P1\�k�QSRgin?hGk

hihG`CXZa�QSR[XZi��ZQOhGkĸ�� `CqOoNi

c = c′¸G\pqS�.`

g1g2 . . . gs = h1h2 . . . hs.

® QOX��.R[tGR[h°t?QSR[X]\pnG�gr|h1h2 . . . hs

¸&n?hGk�kG\grQOXZX.R�k�n?��QO�/R[qDQS�.i@¸ � `CqOoNi

gr | hj� ?X�R[QOh

j ∈ {1, 2, . . . , s} M� \Cl.t¯Ç�a�hjn?a�abR[XZQSR[X.R[h m�VCh?hGR[h ��QOX{`N© k?�

j = s\ph?hGR[t?abR[h<M¤P1\�\p� R[X

hs\pnGl.t¯QOX.R[X.k�n?��QO� R[q�n?hGk

hG`CXZa�QSR[XZiWQS�.i@¸���`CqOoNigr = hs

M?�WnG�Zi.R[QOqSR[h�qOQSR ��R[XZi!hFn?h

g1g2 . . . gr−1 = h1h2 . . . hs−1.

P�n?X.l.t�£ hGk�n?m_iZQS`Ch�h&\Cl.t¾k?R[a ��X]\Ck���`Chf��`CqOoNi�k�QSR1© R[t&\pn?Y?iZn?h?oGM ½�gIP

z{0 ED´ 2CÁ 8_EJµ 2 8jACB+E Á ED´6y8�E+7 4 7 0 E � �O³J5 ¼_R[Q

h ∈ K[X]MG®¯QOX!�.l]t?X.R[QO� R[h

f ≡ g mod h,

�¤R[h?hh | f − g ¸G`_k?R[X@¸�� \C��kG\C�.�.R[qO�/R QS�Zi�L

f ≡ g mod h ⇔ (f mod h) = (g mod h)

! 8F79> } 798F;98 � � ³ �

• X ≡ 0 mod X¸

X + 1 ≡ 1 mod X¸

3X2 + 17X + 5 ≡ 5 mod X¸

f(X) ≡ f(0) mod XM

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5 � ³ � 6 ��� � ��� ����� " �{���¾À

• ��qOqOo�R[abR[QOhGR[X�L&·DR[QOqOn?h?o�a�QOiW��R��Zi qOQSR ��R[XZi f = q(X − α) + f(α)¸&\pqS�]`

f(X) ≡ f(α) mod (X − α)

• (1 +X)10 ≡ 1 + 10X mod X2

• X ·X ≡ −1 mod (X2 + 1)

= 4 µ � �O³ � ��53/ : &�5 � mod h

(H" + &�(*5'&���;( > &�5 � : &�>� + ( � 5 @ �! #"%$'&�(*),+ 6 &."0/�(*>?+

• f ≡ f mod h 6

• f ≡ g mod h⇒ g ≡ f mod h 6

• f ≡ g mod h 6 g ≡ k mod h⇒ f ≡ k mod h @6y8�E+7 4 7 0 E � �O³³ ¼_R[Q

h ∈ K[X]¸h 6= 0

M?P�QSR�¾R[h?o�R

[f ] := f +K[X]h = {f + q · h | q ∈ K[X]}tGR[Q�Ni � $ )��F# ��;)L) $���`Ch

f�/R[� ?oNqOQSl.t

hM

= 4 µ � �O³ ¶ � 8;: ��F�&�( � � > �N5 ��� & /�(*>?+

[f ] = [g] ⇔ f ≡ g mod h (⇔ (f mod h) = (g mod h))

DE&GF�&�(H" @ =⇒ L[f ] = [g]⇒ g ∈ [f ]⇒ h | g − f ⇒ f ≡ g mod h

M ⇐=Lf ≡ g mod h⇒ f = g + ph

� ?XIR[QOhp ∈ K[X]

Mj¼_`Ca�QOif+qh = g+(p+q)h

¸�\pqS�.`[f ] ⊂ [g]

Mj�Wh&\pqS`Co�� `CqOoNi[g] ⊂ [f ]

¸�\pqS�.`[f ] = [g]

M½�gIP

6y8�E+7 4 7 0 E � �O³ K ¼_R[Qh ∈ K[X]

¸h 6= 0

a�QOiGrad(h) = n

M

K[X]/(h) := {[f ] : f ∈ K[X]} ≈ K[X]<n

6y8�E+7 4 7 0 E � �O³<à ® QOX!k?R�¡Gh?QSR[X.R[h¾\pn���k?R[Xa°R[h?o�RK[X]/(h)

���¤R[Q<¥IR[XZm_h! ?Y���n?h?o�R[h�L

[f ] + [g] := [f + g],

[f ] · [g] := [fg].

= 4 µ � �O³�� �B( &." & �'& : 4 5 8 ��7 ;53/N&�5 "W(*5 < F�$N> < &��N5N( & : + 6 ;5 < k[X]

F ( : < < � (?+ � &�(*5'& � 4�����

N+2 + ( �#&�5 � (*53/

�(?+ � ;>*> &�> & � &�5�+ [0]

;5 < ��(*5," &�> & � &�5 + [1] @

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� ��5 � 8F7�; Á EJ´ � 7 4b� 8F> 4 � ��0 ; � E 0� ~ 7 �J79>�7 0 E 5 � ¶

gI�JoNQO�?iDR[QOhGR!L��]L¤© R[��QSR[tFn?h?o�����QS�]l.tGR[h�k?R[X�°R[h?o�R¦k?R[XD�WR��.iZmFq�\C�.�]R[h1n?hGkUk?R[X8¾R[h?o�R¦k?R[XJ��R��Zi.RK[X]<n

MP�QSR�`C�/R[h�k?R�¡Gh?QSR[XZi.R1��k?k�QOiZQS`Ch�n?hGk$°n?qOiZQOY?qOQOmC\piZQS`Ch���`Chy�WR��.iZmFq�\C�.�]R[h�qOQSR � R[XZi �]`Ca�QOi R[QOhGR���k?k�QOiZQS`Chyn?hGk°n?qOiZQOY?qOQOmC\piZQS`Ch°��`Ch��WR��Zi.R[h�L

6y8�E+7 4 7 0 E � �O³ � � ?Xf, g ∈ K[X]<n

�.R[iZ�@R[h���QOX

f +h g := (f + g mod h),

f ·h g := (f · g mod h).

® QOX��¤R[X.k?R[h k?R[h £�hGk?R�^h\p�/R[X � \C�Zi{QOa�abR[X��¤R[oNq�\C�]�.R[h¯n?hGk¯hjn?X�kG\ph?h \pnG�]�.l.t?X.R[QO�/R[h<¸¦� R[h?h �.`ChG�Zi

�h?mFq�\pXZtGR[QOi.R[h�R[hFi.�.R[tGR[h<M

! 8F79> } 798F;98 � �S¶$�

• K[X]/(X − α) ≈ K; [f ] 7→ f(α)

• K[X]/(X2) ≈ K ⊕KX ¸�� `C� R[Q X ·X = 0M

• ��qOqOo�R[abR[QOhGR[X K[X]/(X2 + aX + b) ≈ K ⊕KX ¸?�¤`C�/R[Q X ·X = −aX − b M

= 4 µ � ��¶/5h( :W: & < � (21,&�> ⇔ K[X]/(h)

5 ;>*>?+L&�(*> & :W7 : &�( @DE&GF�&�(H" @ £ �Zi h X.R�k�n?��QO� R[q ¸�\pqS�.` h = h1 ·h2

a�QOih1, h2 6= 0

¸j�.`UQS�Zi \pnGl]t[h1], [h2] 6= 0

¸�\p� R[X[h1] · [h2] =

[h] = 0�&\pqS�]`�t&\pi

K[X]/(h)QOh�k�QSR��.R[a �\pqOq ��n?qOqOi.R[QOqSR[X@M

£ �Zi!t?QOh?o�R[o�R[hhQOXZX.R�k�n?��QO�/R[q ¸ n?hGk��ZQOhGk

[g1], [g2] ∈ K[X]/(h)a�QOi

[g1] · [g2] = 0¸&�]`bQS�Zi�\pqS�]`

g1g2 = 0

mod h¸�kG\C�¤tGR[Q�Ni

h | g1g2MFP1\

hQOXZX.R�k�n?��QO�/R[q QS�Zi@¸ � `CqOoNi

h | g1`Fk?R[X

h | g2¸F\pqS�.`

[g1] = 0`Fk?R[X

[g2] = 0M

½�gIP

= 4 µ � ��¶ �h( :W: & < � (21,&�> ⇒ K[X]/(h)

(H" + &�(*5���� :�� & :G@DE&GF�&�(H" @ ¼_R[Q f ∈ K[X]

a�QOi[f ] 6= 0

M?® QOXWa G�]�.R[h��@R[QOo�R[h<¸kG\C�.�[f ]

R[QOhGRU£ hj��R[X.�]Rc�/R��ZQOiZ��i@M&P1\[f ] 6= 0

¸oNQOqOi

h - fM�P1\

hQOXZX.R�k�n?��QO� R[qDQS�Zi@¸���`CqOoNi

ggT (h, f) = 1MG��qS�.`�oNQO�?iWR���QOh

K[X]R[QOhGRcP1\pX.�Zi.R[qOqOn?h?o

1 = A · f +B · h.

P1\pX]\pnG� � `CqOoNiW\p� R[X[1] = [A] · [f ] + [B] · [h] = [A] · [f ]

QOhK[X]/(h)

M ½�gIP

= 4 µ � ��¶ � ��&�(h ∈ Fp[X]

( :W: & < � (21,&�> ��� � : < d 6 ;5 < " &�(p&�(*5'& � : ( � � $N> @ �! 5N5 (H" +

Fp[X]/(h)&�(*5���� :�� & : � (?+ pd ��> & � &�5 +L&�5 @

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5 � K � 6 ��� � ��� ����� " �{���¾À

DE&GF�&�(H" @ Fp[X]/(h) ' Fp + Fp ·X + . . .+ Fp ·Xd−1 M ½�gIP

= 4 µ � ��¶?³ � ���z�� � ��&�(p&�(*5'& � : ( � � $N> ;5 < d ∈ N @ D (H"B 7 � " ����� :�� $N(H" � " &� (H"G+ ( & : +�/N&�5 R[QOh

� � :�� & : � (?+ pd ��> & � &�5 +L&�5 @ � ( : 1,&��;&�(�� $N5'&�5 (*$N5�(?+

Fpd @

! 8F79> } 798F;98 � �S¶¶L��

F4 ' F2[X]/(X2 +X + 1) = {0, 1, X, 1 + x} �?R���oNQOqOi X2 = −(1 +X) = 1 +XM

e �F8 ' F2[X]/(X3 +X + 1)

Mw �

F9 ' F3[X]/(X2 + 1)M

= 4 µ � ��¶GK � � BD7�ED8j>�79>@ACBD8 2�� 8F> 4 >@= 4 µ�� ��&�( &�5h1, h2 ∈ K[X] �

(?+ggT (h1, h2) = 1 @ �! 5N5 /�(*>?+

K[X]/(h1h2)'−→ K[X]/(h1)×K[X]/(h2).

� ��� ���� ���� � �#� ��� � � ��� �

P�QSR�oj\ph?�@R[hur/\pt?qSR[hZn?hGk �+`Cq��FhG`Ca�XZQOh?o�R

K[X]��R[XZt&\pqOi.R[h �ZQSl]t �]R[t?X �pt?h?qOQSl.t�� QOh¯� R[QSk?R[h¯oNQO�?i�R��

·JR[QOqOn?h?o�a�QOiU�WR��Zi@¸ n?hGksa�\phºmN\ph?h�k?R[h g�n?m_qOQSk�QS�.l.tGR[h �WqOo�`CXZQOiZt?a�nG�Uk�n?X.l.t�� ?t?X.R[h<M<® QOX1�¤`CqOqSR[h k�QSR��.Rg�QOo�R[hG�.l]t&\ �9i�oj\ph?��\pqOqOo�R[abR[QOh�n?hFi.R[X.�ZnGl.tGR[h<M

6y8�E+7 4 7 0 E � ��¶/à g�QOh�hFn?qOqOi.R[QOqSR[X��9X.R[QSR[X �!QOh?oRtGR[Q�Ni ��# � �X�"�7) ��� $�� � � �N¸�� R[h?h°R��!R[QOhGR1���?�?QOqSk�n?h?o

δ : R −→ N0

oNQO�?i@¸G�.`�kG\C�]�!R�� � ?XW\pqOqSRa, b ∈ R ¸ b 6= 0

g�qSR[abR[hji.Rq, r ∈ R oNQO�?i!a�QOi

a = q · b+ rn?hGk

δ(r) < δ(b).

! 8F79> } 798F;98 � �S¶$�L��

Za�QOi

δ(a) = |a| Me �K[X]

a�QOiδ(f) = 2Grad(f) n?hGk δ(0) = 0

�2−∞ = 0

��Mw �

Z[i]a�QOi

δ(a+ bi) = a2 + b2M

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� � � � Á :<;97 ~ 79>@ApBD8 � 7�EJ´8 5 � Ã

� 7´8FED>@ACBJ= � 4 8FE ��Á :<;�7 ~ 79>@ACBJ8 2�� 79ED´&8 �• £ h�g�n?m_qOQSk�QS�.l.tGR[h¾�!QOh?o�R[h�oNQO�?iWR��WR[QOhGR[h �*# ���X�"� ) ��� $�4. � � !�� ��� ��%'��) L

a1 = q1a2 + a3

a2 = q2a3 + a4MMM

ak−1 = qk−1ak + 0

a�QOiδ(a2) > δ(a3) > δ(a4) > . . .

M

• r� R[QDg�qSR[abR[hFi.R a n?hGk b �/R��ZQOiZ�@R[h�R[QOhGR[h ggT M

• gI�WoNQOqOi!k?R[X ggT ��P1\pX.�Zi.R[qOqOn?h?o��]�]\piZ� L

ggT (a, b) = A · a+B · b

• ��R�k?R�� g�qSR[abR[hFi 6= 0QS�Zi�\pn��¦R[QOhGk?R[n?iZQOo�R�®�R[QS�.RUkG\pX.�.i.R[qOqO�&\pX \pqS����X.`_k�n?mFi�QOXZX.R�k�n?��QO�?qSR[X gIqSR[abR[hji.R

��R[QOhGk?R[n?iZQOo��?QS�c\pn�� �+R[XZaUn?i]\piZQS`Ch k?R[X QOXZX.R�k�n?��QO�?qSR[h gIqSR[abR[hji.Rbn?hGk °n?qOiZQOY?qOQOmC\piZQS`Ch a�QOi gIQOh?tGR[Q �i.R[h���M

! 8F79> } 798F; � ��¶��

• 10 = 2 · 5 QOh ZM

• 10 = (1 + i) · (1− i) · (2 + i) · (2− i) QOh Z[i]M

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5 � � � ���y� �����[� 6 ��� z��¤�y+Ä���E!���� 6 ���sÀ

� � ��'H��(/,.' ')(�� .�!$$&'�� ��,] ��� �

6y8�E+7 4 7 0 E � ��5 g�QOhGR % $����$ �(� ��R[QOhGR[XB°R[h?o�RXQS�ZiWR[QOhGR1P1\pX.�Zi.R[qOqOn?h?o

X =⋃

i∈YXi

a�QOi �9Q �Xi 6= ∅

¸ �9QOQ �Xi ∩Xj = ∅ ¸?�¤R[h?h i 6= j

M

X

X4

X2

1X3X

! 8F79> } 798F;98 � � �

• ¼_R[Q X k�QSR °R[h?o�R�\pqOqSR[XB¾R[hG�.l.tGR[h<¸Y = {M,F} ¸ XM =

°R[h?o�R�\pqOqSR[XB �ph?hGR[X@¸XF =

¾R[h?o�R\pqOqSR[X��?X]\pnGR[h<M&P1\ph?h�QS�Zi

X = XM ∪XFR[QOhGR�r R[XZqSR[oNn?h?oGM

• ¼_R[Q X k�QSR ¾R[h?o�RU\pqOqSR[XW��n?i.`N�@¸Y = { X.`Ci , oNX ?h , . . .} k�QSR�°R[h?o�RU\pqOqSR[X��&\pXZ� R[h��

X = X � ��� ∪X ����� ∪ . . .

• X = Z¸Y = {0, 1, 2, . . . , n− 1} k�QSR�¾R[h?o�Rck?R[X!��R��Zi.R

mod nM

Z = [0] ∪ [1] ∪ . . . [n− 1]

6y8�E+7 4 7 0 E � � �

• P�QSR�¾R[h?o�R[hXi

hGR[h?hGR[h���QOX!k�QSR � ��;)L) $�{k?R[X�r R[XZqSR[oNn?h?oGM

• P�QSR;¾R[h?o�RYtGR[Q�Ni � �;# � !��$% $� � $U`_k?R[X � � !���� $� � $��% $� �'$[MJP�QSR��.R�°R[h?o�R�Y&\pX]\pabR[iZXZQS�ZQSR[XZi�k�QSR

��R[X.�]l.t?QSR�k?R[hGR[hsT q�\C�.�.R[h<M

• g�QOh xi ∈ XitGR[Q�Ni � $�5#���;) $� � �" �<k?R[XWT q�\C�.�.R

XiM

• g�QOhGRU�WnG�Z� \pt?q {xi}i∈Y ��`Ch��WR[Y?X�C�]R[hji]\phji.R[hxi ∈ Xi

hGR[h?hjiWa�\ph � $�5 � �;) $� � �" � $ ) � )�� $ %�M

• P�QSR��W�?�?QOqSk�n?h?o π : X −→ Y ; x ∈ Xi 7→ itGR[Q�Ni3#&�" !" �7) ��� $/. !V!�� �R�(�� ��k?R[X ¾R[h?o�R�\pn�� k�QSR

½�nG`CiZQSR[hFi.R[h?abR[h?o�RNM

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5 � �

• P�QSRc��R[q�\piZQS`Chx ∼ y ⇔ ∃i ∈ Y : x ∈ Xi, y ∈ Xi

tGR[Q�Ni���� ��� � �"�$ � � $��������!� �'$�� % $ � �($ �(�� �NM

= 4 µ � �O³ �E( & �!&�>� + (�5 ∼ (H" + &�(*5'&��

�;( > &�5 � : &�>� + ( � 5 6�< @ $ @ &."0/�(*>?+�-� ( � ∼ (H" + X.R��&R�^�QO� -

x ∼ x 798;: >*> & x ∈ X @� (*( � ∼ (H" + � �Fa�abR[iZXZQS�.l]t -x ∼ y ⇒ y ∼ x @� (*(*( � ∼ (H" + iZX]\phG�.QOiZQO� -

x ∼ y 6 y ∼ z ⇒ x ∼ z @DE&GF�&�(H" @ ¼_l.t?X.R[QO�/R[hs¼FQSRck�QSR��W\pqS�a W�?n?h?o��]\pn���oj\p�/R1o�R[h&\pn°\pnG�@M ½�gIPÇ�a�o�R[m�R[t?XZi!� R[m�`Ca�a�iWa�\ph°\pnG�WR[QOhGR[X!� R[qOQSR[�?QOo�R[h � �jn?QO�N\pqSR[h?��X.R[q�\piZQS`Ch°R[QOhGR�r R[XZqSR[oNn?h?o���`Ch

XL

6y8�E+7 4 7 0 E � ��¶ P�QSR ¾R[h?o�R[a] := {b ∈ X | b ∼ a} ⊂ X

tGR[Q�Ni�� � � � � �"�($� � # ��;)L) $!��`Ch a ∈ X M

= 4 µ � �SK �E( & ��;( > &�5 � 4 >� #"�" &�5 1#(*> < &�5 &�(*5'& ��& : > &W/ ;53/ � 5 X @

DE&GF�&�(H" @ ��R�k?R��¦gIqSR[abR[hji QS�ZiIQOh�R[QOhGR[X � �jn?QO�N\pqSR[h?��mFq�\C�]�.R!R[hjiZt&\pqOi.R[h<¸F�¤R[QOq a ∈ [a]Mj£ hG�Z� R��]`ChGk?R[X.R!�ZQOhGkbk�QSR

T q�\C�.�.R[hyh?QSl]tji�qSR�R[X@M?P�R[X!�.Y?XZQOh?o�R[hGk?RU��n?h?m_i�QS�Zi!k�QSR1·D\pi.�²\Cl.tGRN¸&kG\C�]�����¤R[Q<��R[X.�]l.t?QSR�k?R[hGRUT q�\C�]�.R[h�R[QOhGR[hqSR�R[X.R[hsP�n?X.l]tG�.l.t?h?QOiZi1t&\p�/R[h<M £ �Zi�h��pa�qOQSl.t

c ∈ [a] ∩ [b]¸ �.`bQS�Zi

c ∼ an?hGk

c ∼ b¸ �.`Ca�QOiWQS�Zi

a ∼ bM&£ �Zi

d ∈ [a]¸F�.`UQS�Zi

d ∼ a n?hGk�\pqS�]`�\pnGl.t d ∼ b ¸FkÄM t<M d ∈ [b]M_�WqS�.`UQS�Zi

[a] ⊂ [b]M_P�QSR!n?a�o�R[m�R[t?XZi.R £�h?m_qOnG�ZQS`Ch

� `CqOoNiW\pnG��¼ �_a�abR[iZXZQSR[oNX ?hGk?R[h<¸ \pqS�.`[a] = [b]

M ½�gIP

! 8F79> } 798F; � �9à £ hbk?R[X ¼_l]tjn?qSR�qSR[XZhGR[h��ZQSR��.l]tG`Ch�a�QOi � �Fn?QO�C\pqSR[h?��m_q�\C�.�.R[h�n?a���n?o�R[tGR[h<M_¼_R[QX = Z× (Z\

{0}) MG® QOX!k?R�¡Gh?QSR[X.R[h(a, b) ∼ (c, d) ⇔ ad = bc

��qS�.`(1, 3) ∼ (2, 6) ∼ (24, 72)

P�QSR� ?�?qOQSl.tGR ��`Ci]\piZQS`Ch�k?R[X � �jn?QO�N\pqSR[h?��mFq�\C�]�.R[h � ?X!k�QSR��.R��!© R[QS�ZY?QSR[q L

a

b:= [(a, b)]

��qS�.`��ZQOhGky© X Gl.tGR � �jn?QO�N\pqSR[h?��mFq�\C�]�.R[h �1

3=

2

6=

24

72

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5F³ � � ���y� �����[� 6 ��� z��¤�y+Ä���E!���� 6 ���sÀ

P�QSR °R[h?o�R�\pqOqSR[XWT q�\C�.�.R[h�QS�.i�o�R[X]\Ck?Rck?R[XWTWVCXZY/R[XQM

P�QSR��.R TW`ChG�ZiZXZn?m_iZQS`Ch�mN\ph?hya�\phyY?X.`C�?qSR[a�qS`N����R[X]\pqOqOo�R[abR[QOhGR[XZh<¸GQOhGk?R[a a�\phZk�n?X.l]t�R[QOhGR[hy� R[qOQSR[�?QOo�R[h

hjn?qOqOi.R[QOqSR[X���X.R[QSR[hs�!QOh?oRR[X.�.R[iZ��i@MGP1\C�!g�XZo�R[�?h?QS�WQS�Zi!kG\ph?h�R[QOh�TWVCXZY/R[X

Q(R)¸ � � !���� $� � $� # "��5 $��

Q(R)��`ChRo�R[h&\ph?hji@M/¼_`�QS�Zi

Q(k[X]) =: K(X)k?R[X!TWVCXZY/R[X�k?R[X ��������!� �"�($� �"�� # ����!"#$�bQOh

X¸���M © M

1 + 5X + 31X4

3 + 12X +X2∈ R(X).

� ��� � �������� ���� ���;���¼_R[Q

GR[QOhGR�� XZn?Y?Y R1n?hGk

HR[QOhGR1�hji.R[XZoNXZn?Y?Y/Rc��`Ch

GMG®¯QOX!�.l]t?X.R[QO� R[h¾\pnGl.t

H < GM

6y8�E+7 4 7 0 E � � � g�QOhGR � � #+)�#$+! $��# ���()L) $�`Fk?R[X)m_n?XZ� � $+! $��# ���()L) $I��`ChHQS�Zi)R[QOhGR�·JR[QOqOabR[h?o�R!k?R[X��G`CXZa

aH := {ah | h ∈ H} ⊂ G,�¤`C�/R[Q

a ∈ G MGP�QSR °R[h?o�R�\pqOqSR[X ��R[�/R[h?m_q�\C�.�.R[hyhG`CiZQSR[X.R[h���QOX G/H ¸?o�R��ZY?X.`_l.tGR[h��Gab`_k�n?qS`

H� �

G/H := {aH | a ∈ G}P�QSRUr/\pt?qJk?R[X ��R[� R[h?m_q�\C�.�.R[h

[G : H] := |G/H| ∈ N ∪ {∞}hGR[h?hji�a�\ph�k?R[h � ��'$ �1��`Ch

HQOhGM

! 8F79> } 798F;98 � ���

• G = Z¸H = nZ

M�P�QSR ��R[� R[h?m_q�\C�.�.R[h��.QOhGk�o�R[h&\pn�k�QSRc�WR��.iZmFq�\C�.�]R[h

a+ nZ, a = 0, 1, 2 . . . , n− 1.

P�QSR�¾R[h?o�R1k?R[X ��R[� R[h?m_q�\C�.�.R[hyQS�.i!k�QSRU\pqOiZ� R[mN\ph?hji.R�°R[h?o�RZ/nZ

�?k?R[X!£ hGk?R�^�QS�Zi[Z : nZ] = n

M

• G = R2 a�QOiW��k?k�QOiZQS`Ch +�9¥IR[mFi.`CX]\Ck?k�QOiZQS`Ch���M/¼_R[Q

H = {(x, 0); x ∈ R} M?P�QSR ��R[� R[h?m_q�\C�.�.R[hy��`ChH�ZQOhGk�k�QSR ��n?X

x����l.tG�.RcY&\pX]\pqOqSR[qSR[hs� R[X]\Ck?R[h<M

• P�QSR1��`Ch (12)R[XZ�@R[n?oNi.R�ÇWhFi.R[XZoNXZn?Y?Y R

H = 〈(12)〉 = {id, (12)} QOh S3t&\piWk�X.R[Q ��R[�/R[h?mFq�\C�.�]R[h�L

eH = (12)H = {id, (12)},(123)H = (13)H = {(123), (13)},(132)H = (23)H = {(132), (23)}.

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� ��5 � 8jHI8_EJ:<;9=?>@>@8_E 5F³J5

= 4 µ � ��5��� ( � �B( & ��(*5N43"W5'& 1,&�5N4 >� #"�" &�5 1#(*> < &�5 &�(*5'& ��& : > &W/ ;53/ � 5 G @� (*( � � F (H" �L$'&�5 ��F�&�( ��(*5N43"W5'& 1N&�5N4 >� #"�" &�5 /�(213+%&."=&�(*5'&ED ( � &�4'+ ( � 5 @DE&GF�&�(H" @ �9Q � ®¯QOX!k?R�¡Gh?QSR[X.R[h¾R[QOhGRc�WR[q�\piZQS`Ch ∼ \pn��

G¸�QOhGk?R[a ��QOX ��R��.iZqSR[o�R[h�L

a ∼ b ⇔ ba−1 ∈ H

∼ QS�Zi¤R[QOhGR � �Fn?QO�C\pqSR[h?��X.R[q�\piZQS`Ch�L a ∼ a ¸_kG\ aa−1 = e ∈ H M_£ �Zi a ∼ b ¸_�.`�QS�Zi ba−1 ∈ H M_P1\ph?h{QS�Zi�\p�/R[X\pnGl.t(ba−1)−1 ∈ H ¸GkÄM t<M ab−1 ∈ H n?hGk��]`Ca�QOi

b ∼ aM&£ �Zi

a ∼ bn?hGk

b ∼ c¸G�.`��ZQOhGk

ba−1 n?hGk cb−1

g�qSR[abR[hFi.R1��`ChHM�P1\ph?h�\p� R[X!\pnGl]t

ca−1 = cb−1ba−1 ∈ H ¸�kÄM t<M a ∼ c M?P�QSR � �jn?QO�N\pqSR[h?��mFq�\C�]�.R ��`Ch aQS�Zi �'$ �(��k�QSRc�ÄQOh?m��ZhGR[�/R[h?mFq�\C�.�]RN¸?�¤R[qSl]tGRaR[hFiZt��pqOi@M

[a] = {b : b ∼ a} = {b : ba−1 ∈ H} = aH

P�QSR ��R[�/R[h?m_q�\C�.�.R[hy�?QOqSk?R[h¾\pqS�]`�R[QOhGR�r R[XZqSR[oNn?h?o���`ChGM

�9QOQ � P�QSR °n?qOiZQOY?qOQOmC\piZQS`Ch°a�QOiba−1 qOQSR � R[XZiWR[QOhGRc���?�?QOqSk�n?h?o

ba−1· : aH −→ bH, ah 7→ (ba−1)ah = bh.

°n?qOiZQOY?qOQOmC\piZQS`Ch°a�QOiab−1 qOQSR � R[XZiWk�QSR1£�hF��R[X.�.RN¸&\pqS�.`bQS�Zi!k�QSR1�W�?�?QOqSk�n?h?o�R[QOhGRc© Q �#R[m_iZQS`Ch<M ½�gIP

= 4 µ � ��5&5 � = 4 µ � 0 E ] � � N �D� � � � � � ��&�( G &�(*5'& &�5 < >*(��L$'& � : � � & ;5 < H < G&�(*5'& ��5�+L& : / : � � & @

�! 5N5 /�(*>?+

|G| = [G : H] · |H|.�B( & � : < ;53/ &�(*5'& : ��5 +L& : / : � � &E(H"G+� >H" � &�(*5 ��&�(*> & :!< & : � : < 5 ;53/ � 5 G @DE&GF�&�(H" @ G ��QOX.ky�@R[XZqSR[oNi!QOh

[G : H]��R[� R[h?m_q�\C�.�.R[h<¸?k�QSRU\pqOqSR |H| g�qSR[abR[hji.R��/R��ZQOiZ�@R[h � ½�gIP

a7a2

a1

a3

a4

a5

a6

a H2

a H3

a H4 a H5

a H6

a H7

a H1

e

H��`N�.R[Y?t����J`Cn?QS���J\poNX]\ph?o�R

�-� � P SX:��D9��� P &;<MS � JZOLI�J$Y#O$: P ∗ � �� ZRVP † �$����

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5F³ � � ���y� �����[� 6 ��� z��¤�y+Ä���E!���� 6 ���sÀ

z{0&2p0 ;�;9= 2 � ��5 � ��&�(G

&�(*5'& &�5 < >*(��L$'& � : � � & 6 x ∈ G @ �! 5N5 /�(*>?+

ord(x) | |G|.� 5,"G1N&."

�5 < & : &%/�(*>?+

x|G| = e.DE&GF�&�(H" @ gI��QS�Zi ord(g) = | < g > | M ½�g¦P! 8 � 8 2 : Á ED´ � ®�R[h?h a�\ph kG\C�

aQOa�abR[Xyh&\Cl.t X.R�l]tji.�y�Zi]\piZi�h&\Cl]t qOQOh?m_���.l]t?X.R[QO�?i@¸ � R[m�`Ca�a�iya�\ph

� $�� � �F) #$V! $� # ��;)$) $�&MF®�R[h?hbk�QSR!��XZn?Y?Y/R!\p� R[qS�.l]t�QS�Zi@¸pQS�Zi�R���R[oj\pq ¸N`C��a�\phbkG\C�ah&\Cl.tbqOQOh?m���`_k?R[X+X.R�l]tji.�

�.l]t?X.R[QO�?i@¸In?hGk a�\ph �.Y?XZQSl.tji�R[QOh�� \Cl]t ��`Ch � $+! $��# ���()L) $ &¸¤`Ct?hGR��WR�l.tFi.�#��`_k?R[X��ÄQOh?m��#�]M)® QOX��ZY?X.R�l]tGR[h\pnGl.t��/R[QGh?QSl.tFi¤m�`Ca�a�n?i]\piZQO��R[h°��XZn?Y?Y R[h�t��pn�¡Go�hFn?X¦��`Ch ��R[�/R[h?mFq�\C�.�]R[h��jkG\pa�QOi¤�.QOhGkbkG\ph?h{QOa�abR[X k�QSR�ÄQOh?m��ZhGR[�/R[h?mFq�\C�.�]R[h�o�R[abR[QOhji@M

� ��� �������#��� ��������� � � � ������� � � � ����� ���� � �' � � ��� � � � ���6y8�E+7 4 7 0 E � ��5 � £ �Zi

GR[QOhGR���XZn?Y?Y/RN¸?n?hGky�ZQOhGk

g, h,∈ G ¸G�.`�tGR[Q�Ni�kG\C�!g�qSR[abR[hji

ghg−1

kG\C� � !"�� � �"��$ ��� $ � !� h % � � g M

! 8F79> } 798F;98 � �95_³

• £ h�R[QOhGR[X�\p�/R[qS�.l.tGR[hº��XZn?Y?Y R QS�.i!k�QSR��!h?QSl.tFiWQOhFi.R[X.R��.�]\phji@¸kG\ ghg−1 = gg−1h = eh = hM

• £ h S5� R[iZX]\Cl]tji.R

h = (123)n?hGk

g = (15)M+P1\ph?h QS�Zi

ghg−1 = (15)(123)(15) = (523)M)P1\C�

m�`Ch �#n?oNQSR[XZi.RUgIqSR[abR[hjiWQS�.i���QSR�k?R[X�R[QOh�w�� r��_m�R[q ¸&\p� R[X�k�QSRUr/\pt?q1��n?X.k?RcR[X.�.R[iZ��iWk�n?X.l]t

5M

• P�QSR�k?R[XZoNXZn?Y?Y/R D2n = 〈σ, τ〉 M P \p�/R[Q�QS�Zi σ =P�X.R[tjn?h?o n?a

2π/n¸τ =

¼FY?QSR[o�R[qOn?h?o \ph k?R[Xx����l.tG�.RNMP�QSRU� XZn?Y?Y R

D2nQS�Zi!k�QSR�¼ �_a�abR[iZXZQSR[oNXZn?Y?Y/RUR[QOhGR��!X.R[o�R[qOa;�&�NQOo�R[h

n��gIl.m��@MGgI�!oNQOqOi�L

σn = e, τ2 = e, τσ = σ−1τ

P1\C�!gIqSR[abR[hjiστσ−1 �/R��.l]t?X.R[QO�?i�R[QOhGR�¼FY?QSR[qOn?h?o{\ph°k?R[XWk�n?X.l.t σ o�R�k�X.R[tFi.R[h°��l]tG�.R �

© R[abR[XZm�RN¸GkG\C�.�gh1h2g

−1 = gh1eh2g−1 = gh1g

−1gh2g−1.

��qS�.`bQS�Zi kG\C� ��X.`_k�n?mFi�m�`Ch � n?oNQSR[XZi.R[X1g�qSR[abR[hFi.R�oNqSR[QSl.tºk?R[a TW`Ch �#n?oNQSR[XZi.R[h�k?R����IX.`Fk�n?m_i.�@Ml�QOi�£�hGk�n?mF�iZQS`Ch�L

ghng−1 = (ghg−1)n¸G\pqS�]`�t&\p� R[h

hn?hGk

ghg−1 �ZQSl]tGR[X�k�QSR oNqSR[QSl.tGR�Æ�X.k�hjn?h?oGM

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� � � z{0 E�� Á ´&= 4 7 0 E � �y0&2C� =G; 4 8_79;98 2�Á E ~ � =?: 4@0&2 ´ 2CÁJ}+} 8 5F³ �

6y8�E+7 4 7 0 E � ��5?¶ ¼_R[QH < G

R[QOhGRcÇ�hji.R[XZoNXZn?Y?Y/Rcn?hGkg ∈ G M?P1\ph?h�QS�Zi

gHg−1 = {ghg−1 | h ∈ H}

R[QOhGRbÇ�hji.R[XZoNXZn?Y?Y/R���`ChGn?hGkºtGR[Q�Ni #&!"�� � ����$�� � $ � � $ � �"� �V5�5 $c��`Ch

HMHtGR[Q�Ni !"� % �"� QOh

G`Fk?R[X

� !��$% �"��� $�� �$�� � !" G ¸��¤R[h?h

gHg−1 = H, ∀g ∈ G.��`Ci]\piZQS`Ch�L

H CGM

© R[abR[XZm�RN¸GkG\C�.�Ho�R[h&\pn°kG\ph?h�hG`CXZa�\pqJQS�Zi@¸��¤R[h?h

gH = Hg

� ?X{\pqOqSRg ∈ G

¸¦kÄM t<M¤k�QSR��ÄQOh?m��#��n?hGk¯�WR�l]tji.�ZhGR[�/R[h?mFq�\C�]�.R[h¯�ZiZQOa�abR[h ?� R[X.R[QOh<M¤£�h¯R[QOhGR[X�\p�/R[qS�.l.tGR[h��XZn?Y?Y/Rc�ZQOhGk�\pqS�.`�h&\pi ?XZqOQSl]t¾\pqOqSRcÇ�hji.R[XZoNXZn?Y?Y/R[h�hG`CXZa�\pq M

= 4 µ � ��5_K ��&�(φ : G −→ G′ &�(*5 � : � � &�5N$ ��������� :�� $N(H" � " @ �! 5N5 (H" +

ker(φ) CG.

DE&GF�&�(H" @ ¼_R[Q H := ker(φ)¸ �.R[Q

g ∈ G ¸&n?hGks�.R[Q a ∈ gHg−1 ¸ \pqS�]` a = ghg−1 a�QOi h ∈ ker(φ)M/P1\ph?h

QS�Ziφ(a) = φ(ghg−1) = φ(g)φ(h)φ(g−1) = φ(g)φ(g−1) = e′,

\pqS�.`�QS�Zia ∈ ker(φ) = H

M?P1\C��tGR[Q�NigHg−1 = H

M ½�gIP

! 8F79> } 798F;98 � �95&Ã

• P1\C��¥I`CXZ�@R[QSl]tGR[hº¼FQOoNh�QS�.i!R[QOh°��XZn?Y?Y R[h?tG`Cab`Cab`CXZY?t?QS�.aUnG�

¼FQOoNh: Sn −→ µ2, σ 7→ ¼FQOoNh

(σ).

P R[XWT1R[XZh�QS�Zi!k�QSRc\pqOi.R[XZh?QSR[X.R[hGk?R�� XZn?Y?Y RN¸&\pqS�]`An C Sn

M

• £ h�k?R[XWP�QSR�k?R[XZoNXZn?Y?Y R G = D2nL

< σ > CG,\p� R[X

< τ > 6CG.

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5F³³ � ���y� �����[� 6 ��� z��¤�y+Ä���E!���� 6 ���sÀ

• V = {1, (12)(34), (13)(24), (14)(23)} < A4��QOX.k k�QSR � bj� R � �$� ) ��� $ � � $�� $�� ���$�V5�5 ${o�R[h&\ph?hFi@M

�\C�.�.R[h ��QOXA4

\pn��U\pqS��� XZn?Y?Y R�\pqOqSR[X�P�X.R[tjn?h?o�R[h k?R���·JR[iZX]\CR�k?R[X.��¸¦�.`º�/R[m�`Ca�abR[h ��QOX�R[QOhGR[h� XZn?Y?Y R[h?tG`Cab`Cab`CXZY?t?QS�Za�nG�

A4 −→ S3,

� R[qSl.tGR[X �#R�k?R[X�P�X.R[tjn?h?o�k�QSR R[hFi.�ZY?X.R�l.tGR[hGk?R��DR[XZa�n?i]\piZQS`Ch°k?R[X!P�Q�\po�`Ch&\pqSR[h���nG`CX.k�hGR[i@M?P R[X!T1R[XZhQS�.i�o�R[h&\pn

V¸G\pqS�]`�QS�Zi

V CA4M

= 4 µ � ��5�� ��&�(H CG

&�(*5 � � :�� >?+L&�(*> & :G@ �! 5N5 (H" + < : � $

aH · bH := abH

&�(*5'& F�$N> < &��N5N( & : +L& �'& : 4 5 8 ��7 ;53/ 7�< & : � &�53/N& < & : � & 1,&�5N4 >� #"�" &�5

G/H/N&W/N& 1N&�5 @

DE&GF�&�(H" @ ® QOXUa G�]�.R[h ��R[XZQ ¡G��QSR[X.R[h<¸)kG\C�.��k�QSR��.R�P R�¡Gh?QOiZQS`Ch h?QSl.tji���`Ch k?R[XU® \pt?q k?R[X��WR[Y?X�C�]R[hji]\phji.R[h\p�?t��ph?oNi@M&£ �Zi

a′H = aHn?hGk

b′H = bH¸?�.`�QS�Zi

a′ = ahn?hGk

b′ = bh′a�QOi

h, h′ ∈ H M?�WqS�.`�QS�.i

a′b′ = ahbh′.

P1\�\p� R[XH C G

¸oNQOqOiHb = bH

M P�QSR�� � R�k?R[n?i.R[i@¸ÄkG\C�]����QOXhb ∈ Hb �.l.t?X.R[QO�/R[hºm�VCh?hGR[h \pqS� bh′′ a�QOi

h′′ ∈ H M?gI� � `CqOoNi a′b′ = abh′′h′n?hGk��.`Ca�QOi

a′b′H = abHM ½�gIP

= 4 µ � ��5�� ��&�(G

&�(*5'& �0: � � & ;5 < H CG&�(*5 � � :�� >?+L&�(*> & :G@� ( � �B( & � &�53/N&

G/H �(?+ < & : � 1,&�5 < &��N5N( & : +L&�5 �'& : 4 5 8 ��7 ;53/ (H" + &�(*5'& � : � � & � /N&�5 5N5 + �&\pm_i.`CXZoNXZn?Y?Y R

� < & : ½�nG`CiZQSR[hji.R[h?oNXZn?Y?Y/R � @� (*( � �B( & 4� 5 � 5N(H" � $'& � 1N1#(*> < ;53/π : G −→ G/H, a 7→ aH

(H" + &�(*5 �0: � � &�5N$ ��������� :�� $N(H" � " � (?+

ker(π) = H.

DE&GF�&�(H" @ �9Q �JP1\C� � \Cl.t?X.R�l]t?hGR[h<¸�kG\C�.� G/H �.`�QOh�k?R[X)·+\pi�R[QOhGR ��XZn?Y?Y/R ��QOX.kĸpQS�Zi+�+qSR[Q�j\pXZ�/R[QOi@M ��R[n?iZX]\pqSR��g�qSR[abR[hFiWQOh

G/HQS�Zi

eH¸?£�hF��R[X.�.R1��`Ch

aHQS�.i

a−1HM

�9QOQ �π(ab) = abH = aH ·bH = π(a)·π(b)

¸F\pqS�.`WQS�.iπR[QOh���XZn?Y?Y/R[h?tG`Cab`Cab`CXZY?t?QS�ZaUnG�@Mj£ �Zi

π(a) = eH¸

�.`�QS�ZiaH = H

¸&\pqS�.`a ∈ H M&¼_`Ca�QOiWQS�Zi ker(φ) = H

M ½�gIP� # ED:=KM<MU=V�9DI�<MSW@�<6J$Y . K6:O<6Y�P ∗ �$����SVP † �9S�"ZQ

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� � � z{0 E�� Á ´&= 4 7 0 E � �y0&2C� =G; 4 8_79;98 2�Á E ~ � =?: 4@0&2 ´ 2CÁJ}+} 8 5F³/¶

= 4 µ � � ��� � �y0� 0� 0&2p} B+798j>@= 4 µ � �D2�À�2CÁJ}+} 8_E � ��&�(G

φ−→ G′ &�(*5 � ��������� :�� $N(H" � " � 5 � : � ��&�5 @ ��&�( H := ker(φ) CG @ �! 5N5 /�(213+0&." &�(*5'&�5 &�(*5 < &�N+ ( / 1,&." + (

���+L&�5 � : � � &�5N$ ��������� :�� $N(H" � "

ψ : G/H −→ G′,

"��< #"�"

φ = ψ ◦ π.�B( &." &."

ψ(H" + (*5 � &�4'+ ( ;5 < >*( & 7 & : +0&�(*5'&�5 � "

����� :�� $N(H" � "

G/H'−→ φ(G).

g�QOhGR[hy�.`CqSl]tGR[hy¼�\Cl]tj��R[XZt&\pqOi!�Zi.R[qOqOi a�\ph�` ��i¦QOh�R[QOhGR[a #&!(%'%'��� ����� � $� � �����"���(%'%«kG\pX@¸FkG\C� tGR[Q�Ni@¸j� R[h?ha�\ph�QOh k�QSR��]R[a P�Q�\poNX]\pa�a \pn�� ���¤R[Q¦®�R[o�R[h ��`Ch R[QOhGR[a ��n?h?m_ic��n R[QOhGR[a¿\phGk?R[X.R[h�o�R[q�\ph?o�R[h mN\ph?h<¸kG\ph?h�Y&\C�.�.QSR[XZi �#R[� R[QOqS�!QOhG�Zo�R��]\pa�i�kG\C��� qSR[QSl.tGRNM

//

π!!DD

DDDD

DDG′

G/H

ψ

<<zzzzzzzz

DE&GF�&�(H" @ ¼_R[Q a ∈ GM)P�QSRy�G`CX.k?R[XZn?h?o

φ = ψ ◦ π R[XZ����QOh?oNiφ(a) = ψ ◦ π(a) = ψ(aH)

M)®�R[h?h k�QSR���?�?QOqSk�n?h?o

ψR�^�QS�ZiZQSR[XZi@¸G�.`�QS�.i!�ZQSR t?QSR[X.k�n?X.l.t¾R[QOhGk?R[n?iZQOo � R��ZiZo�R[qSR[oNi@M?® QOX!�.R[iZ�@R[h���R[X.�ZnGl]tGR[h¾\pqS�.`

ψ(aH) := φ(a)

��ny�.R[iZ�@R[h<M?P�QSR�� QS�Zi=� !&�'�R�'$�� � $�� �TL�£ �Zia′H = aH

¸��.`�QS�Zia′ = ah

� ?X R[QOhh ∈ H M�P1\ph?h�QS�Zi ψ(a′H) =

φ(ah) = φ(a)φ(h) = φ(a) = ψ(aH)¸&�¤R[QOq

φ(h) = e′¸?kG\

H = ker(φ)M

ψQS�.i!QOh �#R[m_iZQO�2L

ψ(aH) = ψ(bH) ⇔ φ(a) = φ(b)⇔ φ(ab−1) = e′ ⇔ ab−1 ∈ H ⇔ a ∈ bH ⇔ aH =

bHP1\

φ = ψ ◦ π n?hGk π �.n?X �#R[m_iZQO� ¸�� `CqOoNi Bild(φ) = Bild(ψ)M ½�gIP

! 8F79> } 798F;98 � � �Ä5

• ¼_R[Q G R[QOhGRU��XZn?Y?Y/R1n?hGkg ∈ G MGP�QSRc���?�?QOqSk�n?h?o

φ : Z −→ G, k 7→ gk

QS�.i!R[QOh¾��XZn?Y?Y/R[h?tG`Cab`Cab`CXZY?t?QS�ZaUnG�@M £ �Ziord(g) = n

¸?�.`�QS�Ziker(φ) = nZ

¸�n?hGky��QOX�� R[m�`Ca�abR[h

ψ : Z/nZ −→ G

a�QOi< g >

\pqS�!© QOqSkÄMG��qS�.`Z/nZ '< g >

M

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5F³ K � ���y� �����[� 6 ��� z��¤�y+Ä���E!���� 6 ���sÀ

• ¼FY/R[��Q�\pq� \pqOqJ��`Ch �9Q � a�QOiG = Z/nZ× Z/mZ

¸g = (1, 1)

L

φ : Z −→ Z/nZ× Z/mZ, k −→ (k mod n, k mod m)

P R[XWT1R[XZh���`Chφ�/R��Zi.R[tFi�\pnG�!k?R[h �ZR[h?QOo�R[h

k ∈ Z¸�� ?X!k�QSR

k ≡ 0 mod n, k ≡ 0 mod m.

£ �ZiggT (m,n) = 1

¸<�.` ��`CqOoNiUt?QSR[X]\pnG�k ≡ 0 mod nm

n?hGkker(φ) = nmZ

M® QOX1�/R[m�`Ca�abR[hkG\ph?h°R[QOhGR[h�QOh �#R[iZQO��R[hs��XZn?Y?Y R[h?tG`Cab`Cab`CXZY?t?QS�.aUnG�

ψ : Z/mnZ ↪→ Z/nZ× Z/mZ.

®�R[QOqD� R[QSk?R���XZn?Y?Y/R[h�oNqSR[QSl.t¾�FQSR[qSR�g�qSR[abR[hji.R��/R��ZQOiZ�@R[h<¸GQS�Zi�k�QSR��W�.`Coj\pX�R[QOh�£ �]`Cab`CXZY?t?QS�ZaUnG�@M P1\C�QS�.i�o�R[h&\pn°k?R[X �¤t?QOhGR��ZQS�.l]tGRU��R��Zi.�]\piZ��M

• Sn/An = µ2M

• D2n/ < σ >= µ2M

• φ : A4 −→ S3¸V = ker(φ)

¸A4/V ≈ A3

M

� ��� � ��� � � � ��� � ��� � ����� ��� ��� ��� �£ �Zi

H CG¸j�.`Um�VCh?hGR[h���QOX¤n?hG�¤��`CX.�Zi.R[qOqSR[h<¸�kG\C�.� k�QSR ��XZn?Y?Y/R

G\pnG�¤���¤R[Q/m_qSR[QOhGR[X.R[h;� ¼Fi Gl]m�R[h\�

Hn?hGk

G/H\pn��9o�R[�&\pn?i!QS�.i@M?� ?X!R[QOhGR�\p� R[qS�.l]tGR���XZn?Y?Y/R

Go�R[qOQOh?oNiWR��W�.`�X.R�l.tji�oNn?i@¸

G\p�?��n?�?X.R�l.tGR[h¾n?hGk�\pqS�

\pnG��m_qSR[QOhGR[X.R[h�·JR[QOqSR[h°��nG�]\pa�abR[h?o�R[�&\pn?i���ny� R[iZX]\Cl]tji.R[h<M

= 4 µ � � � � ��&�(G

&�(*5'& &�5 < >*(�� $'& 31N&�>H" � $'& � : � � & @ �! 5N5 (H"G+G� : � < ;4�+ � 5 &�5 < >*(��L$'&�5 � �34 >*(H" �L$'&�5

� : � � &�5�-

G ' Z/n1Z× Z/n2Z× . . .× Z/nrZ

© R[��`CX!��QOXWk?R[h�© R[�¤R[QS�Wo�R[� R[h<¸G�?X]\pnGl.tGR[h���QOX�R[QOh��<R[a�a�\ L

� 8 � � = � � � � ��&�(G

&�(*5'& &�5 < >*(�� $'& 31,&�>H" � $'& � : � � & 6 h ∈ G&�(*5 ��> &

�&�5 +

� � 3(� > & : � : < 5 ;53/ @��&G+��;&

H := 〈h〉 < G @ ��&�( pi : G −→ G/H < ( & 4� 5 � 5N(H" � $'& � 1N1#(*> < ;53/ 7�< ( & �� 4�+ � : / : � � & @ �! 5N5/�(*>?+�- 798;: >*> & a ∈ G/H /�(213+0&." &�(*5

b ∈ G �(?+π(b) = a

;5 <

ord(a) = ord(b).

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� �O³ � ~ 8j=G;98 Á E ~ � =?: 4@02�2 7�EJ´8 5F³<Ã

DE&GF�&�(H" @ ¼_R[Q N = ord(h) = e(G)M)¼_R[Q

n = ord(a)n?hGk �.R[Q

c ∈ G R[QOh �/R[qOQSR[�?QOo�R[X � � � �)��`Cha¸JkÄM t<M

π(c) = aMGg¦��QS�Zi

π(cn) = π(c)n = an = e ∈ G/H.��qS�.`�QS�Zi

cn ∈ ker(π) = H¸GkG\C��tGR[Q�Ni!��QOX!m�VCh?hGR[h°�.l.t?X.R[QO�/R[h

cn = ht

� ?XWR[QOhtM&gI�WQS�Zi

ord(c) = n′n¸?n?hGk�k�QSR��WQS�Zi�R[QOh°·JR[QOqSR[X���`Ch

NM&��qS�.`

nn′ | N M&��qS�.` e = cnn′

= htn′ ¸

n?hGk��.`Ca�QOiN | tn′ MN£ hG�Zo�R��]\pa�i¤\pqS�.` nn′ | tn′ ¸N`Fk?R[X n | t MC®¯QOX��.l]t?X.R[QO� R[h t = kn

n?hGkb�.R[iZ�@R[hb = ch−k

MgI��QS�.i

bn = cnh−kn = cnht = e.

��qS�.`�QS�Ziord(b)

R[QOh�·DR[QOqSR[XW��`ChnM?P1\�\p�/R[X

π(b) = aÆ�X.k�hjn?h?o

nt&\pi@¸ ��`CqOoNi

ord(b) = nM ½�g¦P

DE&GF�&�(H" @ ��© R[� R[QS�Wk?R�� ¼�\piZ�@R�� �® QOX!� R[hFn?iZ�@R[h¾£ hGk�n?m_iZQS`Ch�h&\Cl.t |G| M&P�QSRU��nG�.�]\po�RU�ZiZQOa�aUi�iZXZQO�FQ�\pqSR[XZ� R[QS�.R � ?XWk�QSR���XZn?Y?Y/Rca�QOi�R[QOhGR[ag�qSR[abR[hFi@M��Wh?o�R[hG`Ca�abR[h<¸�k�QSR¾© R[t&\pn?Y?iZn?h?o QS�Zi{�/R[��QSR��.R[h � ?X�\pqOqSRº\p� R[qS�]l.tGR[h ��XZn?Y?Y/R[h a�QOi

< |G|g�qSR[abR[hFi.R[h<MJ¼_R[QH =< h >

n?hGkhR[QOhsg�qSR[abR[hFi1a�\�^_QOa�\pqSR[XU�X.k�hjn?h?oyQOh

GMl�QOi £�hGk�n?m_iZQS`Ch¾��QS�.�.R[h

��QOX@¸GkG\C�.�G/H =< a1 > × < a2 > × . . .× < ar > .

rn �ZR�k?R[a k?R[Xai¡GhGk?R[h°��QOX!g�qSR[abR[hji.R

bi ∈ Ga�QOi

ord(ai) = ord(bi)M&¼_R[Q

G′ =< b1 > × < b2 > × . . .× < br >⊂ G.© R[abR[XZm�RN¸?kG\C�.�

πk�QSRc��XZn?Y?Y/R

G′ QS�.`Cab`CXZY?t�\pn�� G/H \p�?�?QOqSk?R[i@M?P�R�¡Gh?QSR[X.R1R[QOhGR[h���XZn?Y?Y/R[h?tG`Cab`Cab`CX#�Y?t?QS�Za�nG�

λ : H ×G′ −→ G, (h, g′) 7→ hg′.

s\ph���R[XZQ ¡G��QSR[X.R@Lker(λ) = e

M P1\Gn?hGk

H ×G′ oNqSR[QSl]ts�FQSR[qSR�g�qSR[abR[hji.R��/R��ZQOiZ�@R[h<¸GQS�Zi λ R[QOh¾£ �]`Cab`CX#�Y?t?QS�Za�nG�@MG��qS�.`G =< h > × < b1 > × < b2 > × . . .× < br > .

P�QSR1£�hGk�n?m_iZQS`Ch�QS�Zi���`CqOqS�.i�phGk�QOo � ½�gIP

� ��� � ������� ���� � �' �(�������2��� �6y8�E+7 4 7 0 E � � �G³ ¼_R[Q

RR[QOh��!QOh?oGM?gIQOhGRU·JR[QOqOabR[h?o�R

I ⊂ R tGR[Q�Ni � � $��"� ¸?�¤R[h?hL��a, b ∈ I ⇒ a+ b ∈ I ¸e �a ∈ I, r ∈ R ⇒ ra ∈ I M

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5F³ � � ���y� �����[� 6 ��� z��¤�y+Ä���E!���� 6 ���sÀ

��nG��L���n?hGk¾e � ��`CqOoNi��.` ��`CXZi@¸kG\C�.� \pnGl.ta − b ∈ I ¸&kG\C��tGR[Q�Ni I QS�.i�\pn�� �ZR�k?R[h°�&\pqOq)R[QOhGR�Ç�hji.R[XZoNXZn?Y?Y/R�/R[� ?oNqOQSl.t

+M

! 8F79> } 798F;98 � � �¶

• I = {0} n?hGk I = R�ZQOhGk�iZXZQO�_Q�\pqSR[XZ�¤R[QS�]RU£ k?R@\pqSRNM

• I = nZ = {nk | k ∈ Z} QS�.iWR[QOh�£ k?R@\pqJQOh ZM

• £ �Zi a ∈ R ¸?�.`�QS�Zi(a) := aR = Ra = {ar | r ∈ R}

R[QOh°£ k?R@\pqÄQOhRn?hGky��QOX.kykG\C�W��`Ch

aR[XZ�@R[n?oNi.R � �(�V5#���X�'$��"�?o�R[h&\ph?hFi@M

= 4 µ � � �GK ��&�(φ : R −→ S

&�(*5 � (*53/�$��������� :�� $N(H" � " @ �! 5N5 (H" + < & : � & : 5

ker(φ) := {r ∈ R | φ(r) = 0}&�(*5 � < &� > @DE&GF�&�(H" @ ¼FQOhGk a, b ∈ ker(φ)

¸&�.`bQS�Ziφ(a) = φ(b) = 0

MP1\ph?h°\p� R[X \pnGl.tφ(a+ b) = φ(a) + φ(b) = 0

¸\pqS�.`

a+ b ∈ ker(φ)M£ �Zi

a ∈ ker(φ)n?hGk

r ∈ R ¸G�.` ��`CqOoNiφ(ar) = φ(a)φ(r) = 0 · φ(r) = 0

¸&\pqS�]`bQS�Ziar ∈ ker(φ)

M ½�gIP

6y8�E+7 4 7 0 E � � �/à ¼_R[QRR[QOh��!QOh?oG¸

I ⊂ R R[QOh�£ k?R@\pq M?®¯QOX!�]l.t?X.R[QO�/R[h

a ≡ b mod I

o�R��ZY?X.`_l.tGR[h��aQS�.i�m�`Ch?oNXZnGR[hji

bab`_k�n?qS`

I�²¸�� R[h?h

a− b ∈ I.

= 4 µ � � ��� ��53/ : &�5 � ��� < ;> � I (H" + &�(*5'&�� � ;( > &�5 � : &�>� + ( � 5 @

6y8�E+7 4 7 0 E � � �$�

[a] := a+ I = {a+ b | b ∈ I}tGR[Q�Ni � $$) �F# ���()L) $ � !" a % !+�(���!

IM_gI� QS�Zi o�R[h&\pnyk�QSR � �Fn?QO�N\pqSR[h?��mFq�\C�.�]RW�/R[� ?oNqOQSl.t�k?R[X �WR[q�\piZQS`Ch ��TW`Ch��

oNXZnGR[h?�cab`_k�n?qS`I�²M

6y8�E+7 4 7 0 E � � � �R/I := {[a] | a ∈ R} QS�Zi!k�QSR�¾R[h?o�Rc\pqOqSR[XW��R��ZiZmFq�\C�]�.R[hyab`Fk�n?qS`

IM

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� �O³ � ~ 8j=G;98 Á E ~ � =?: 4@02�2 7�EJ´8 5F³ �

6y8�E+7 4 7 0 E � � � 5 ® QOX!k?R�¡Gh?QSR[X.R[h°R[QOhGRc��k?k�QOiZQS`Ch+n?hGk�R[QOhGR��n?qOiZQOY?qOQOmN\piZQS`Ch · \pn�� R/I L

[r] + [s] := [r + s],

[r] · [s] = [rs].

P�QSR�� QS�Zi � !&�'�R�'$�� ��$ ����1G£ �Zi[r′] = [r]

n?hGk[s′] = [s]

¸?�.`�QS�Zir′ − r ∈ I n?hGk s′ − s ∈ I ¸?kÄM t<M r′ = r + c

¸s′ = s+ d

a�QOic, d ∈ I M P \ph?h¾QS�Zi r′ + s′ = r + s+ c + d

¸ \pqS�.`[r′ + s′] = [r + s]

¸&� R[QOqc + d ∈ I M®�R[QOi.R[X

r′s′ = (r + c)(s+ d) = rs+ cs+ rd+ cd¸&\pqS�.`

[r′s′] = [rs]¸�� R[QOq

cs, rd, cd ∈ I M

= 4 µ � � � �

� ( � �E( & � &�53/N&R/I �

(?+ < &�5 � 1N&�5 < &��N5N( & : +L&�5 � � & : + ( � 5'&�5 +;5 < · (H" +�&�(*5 � (*53/ 6 /N&�5 5N5 + �&\pm_i.`CXZXZQOh?o

1,&�� 8 /�>*(�� $ I @� (*( � �B( & 4� 5 � 5N(H" � $'& � 1N1#(*> < ;53/ π : R −→ R/I(H" + &�(*5 � (*53/�$

��������� :�� $N(H" � " @DE&GF�&�(H" @ g�i�� \C�Wq�\ph?oN��QSR[XZQOoGM&¥IR[XZQ ¡G��QSR[X.R[hs¼FQSRck�QSRc�W^�QS`CabR � ½�gIP

= 4 µ � � �&� � �y0� 0� 0&2p} B+798j>@= 4 µ#� �J2{� 7�ED´&8 � ��&�(φ : R −→ S

&�(*5 � (*53/�$��������� :�� $N(H" � " @ ��&G+��;&

I := ker(φ) @ �! 5N5 /�(213+ &."0/N&�5 &�(*5ψ : R/I −→ S �

(?+

φ = ψ ◦ π.

ψ(H"G+ (*5 � &�4'+ ( ;5 < (*5 < � ( & : + &�(*5'&�5 � (*53/�(H" ����� :�� $N(H" � "

R/I'−→ φ(R).

DE&GF�&�(H" @ ®¯QOX!�ZQOhGk�o�R[����n?h?o�R[h<¸ψ([a]) = φ(a)

��nº�.R[iZ�@R[h<MÄP�QSR�� QS�Zi �¤`Ct?qSk?R�¡Gh?QSR[XZi�n?hGksQOh �#R[m_iZQO� M/��qOqSR�� �9n?h?m_iZQS`Ch?QSR[XZic�.R[t?X��pt?h?qOQSl.t ��QSR�� R[Q¦��XZn?Y?Y/R[h<M½�gIP

6y8�E+7 4 7 0 E � � � ³ ¼_R[QS ⊂ R R[QOhGR¦�/R[qOQSR[�?QOo�R ·JR[QOqOabR[h?o�R ��`Ch

RM�P \C�Dm_qSR[QOhG�Zi.R¦£ k?R@\pq ¸�kG\C�)\pqOqSR¦gIqSR[abR[hji.R

��`ChSR[hFiZt��pqOi@¸&��QOX.k�kG\C�!��`Ch

SR[XZ�@R[n?oNi.R�£ k?R@\pq<o�R[h&\ph?hji@¸���`Ci]\piZQS`Ch

(S)M � �

(S)QS�ZiWk?R[XWP�n?X.l.tG�]l.t?h?QOiZi

\pqOqSR[XW£ k?R@\pqSRN¸?�¤R[qSl]tGRSR[hFiZt&\pqOi.R[h<¸Gn?hGky�/R��Zi.R[tji�\pnG�W\pqOqSR[h $��� ������� $� � � #$ �"�L#"!(% !�� #������!�#$�

r1s1 + r2s2 + . . . + rnsn,

�¤`C�/R[Qri ∈ R

n?hGksi ∈ S

M� ' T O$KHU ��<?@3�D:OI�KM<MY;:BJ I�:=Y G;KMKM: J &DS �V:OI�KM<6YD:BJLI�:OY � K O :��DI�J

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5�¶$� � ���y� �����[� 6 ��� z��¤�y+Ä���E!���� 6 ���sÀ

£ �ZiS = {s1, s2, . . . , sr}

¸&�.`��.l.t?X.R[QO�?i�a�\phyh&\pi ?XZqOQSl.t(s1, s2, . . . , sr)

�.i]\piZi({s1, s2, . . . , sr})

MGg¦��QS�Zi

(s1, s2, . . . , sr) = {r∑

i=1

risi}.

��qOi.R[XZh&\piZQO��R ��`Ci]\piZQS`Ch�L(s1, s2, . . . , sr) = Rs1 +Rs2 + . . . +Rsr

! 8F79> } 798F;98 � � � ¶

• £ h ZoNQOqOi

(6, 8) = (2)M&gI� QS�Zi�h��pa�qOQSl.t

2 = 8− 6¸G\pqS�.`

2 ∈ (6, 8)n?hGky�.`Ca�QOi

(2) ⊂ (6, 8)M&��� R[X

6 = 3 · 2 ¸ 8 = 4 · 2 ¸G\pqS�.` 6 ∈ (2)n?hGk

8 ∈ (2)n?hGk��]`Ca�QOi

(6, 8) ⊂ (2)M&�WqS�.`�oNQOqOi ��qSR[QSl.t?tGR[QOi@M

• (a) ⊂ (b)oNQOqOi!o�R[h&\pn�kG\ph?h<¸?� R[h?h

a ∈ (b)¸&\pqS�.`

a = rb� ?X!R[QOh

r ∈ R M��WqS�.`

(a) ⊂ (b) ⇔ b | a·DR[QOqO�&\pXZm�R[QOi�QS�.iW\pqS�.`�£�h?m_qOnG�ZQS`Chy��`Ch�£ k?R@\pqSR[h<M

• ��qOqOo�R[abR[QOhGR[XWoNQOqOi!QOh ZL

(m,n) = (ggT (n,m))

© R[� R[QS��LF¼_R[Qd := ggT (m,n)

MjgI��oNQO�?iA,B ∈ Z a�QOi

d = Am+Bn¸p\pqS�.`cQS�Zi

d ∈ (m,n)¸N� \C�I��n

(d) ⊂ (m,n)� ?t?XZi@M&P1\

d | m n?hGkd | n ¸GQS�Zi (m) ⊂ (d)

n?hGk(n) ⊂ (d)

¸&�.`�kG\C�.�(m,n) ⊂ (d)

M��qS�.`

(d) = (m,n)M

6y8�E+7 4 7 0 E � � � K g�QOh��WQOh?oRtGR[Q�Ni � �(�V5#���X�'$��"� ��� ��¸G�¤R[h?h �#R�k?R��!£ k?R@\pq

I ⊂ R R[QOh#��\pn?Y?iZQSk?R@\pqDQS�Zi@M

= 4 µ � � � Ã � "G+R

&�(*5 &�;4 >*( < (H" �L$'& : � (*53/ 6 " � (H" + R &�(*5 � �+ ( < &� > : (*53/ @

DE&GF�&�(H" @ £�h�R[QOhGR[a R[n?m_qOQSk�QS�.l]tGR[h°�!QOh?o�QS�.i!R[QOhGRc�?n?h?m_iZQS`Ch

δ : R −→ N0

��`CXZo�R[o�R[�/R[h<¸G�.`�kG\C�.� � ?XW\pqOqSRa ∈ R n?hGk

b ∈ R ¸ b 6= 0¸?R[QOh

qR�^_QS�ZiZQSR[XZi a�QOi

δ(a− qb) < δ(b)MI = (0)QS�Zi�R[QOh � \pn?Y?iZQSk?R@\pq M ��R[t?abR[h¾��QOX1\ph<¸

I 6= (0)M®_�pt?qSR�R[QOh

b ∈ I \ {0} a�QOi δ(b) a�QOh?QOa�\pq M<¼_R[Q a ∈ I�/R[qOQSR[�?QOoGM_g¦� oNQO�?iq ∈ R a�QOi

δ(a− qb) < δ(b)M?P1\�\p�/R[X a�QOi

an?hGk

b\pnGl.t

a− qb QOh I qOQSR[oNi n?hGk δ(b)a�QOh?QOa�\pqJ� \pX@¸?aUnG�]�a− qb = 0

�.R[QOh<¸GkG\C�!tGR[Q�Nia ∈ (b)

MG��qS�.`I = (b)

M ½�gIP

z{0&2p0 ;�;9= 2 � � � �Z 6 K[X]

� � � � :�� & : � ;5 < Z[i]"W(*5 < � � + ( < &� > : (*53/N& @

P R[X��D`Cq��_hG`Ca�XZQOh?oK[X,Y ]

�9n?hGk�\pnGl.tZ[X]

�¦QS�Zi m�R[QOh ��\pn?Y?iZQSk?R@\pqOXZQOh?oVLGkG\C� £ k?R@\pq(X,Y )

mC\ph?h�h?QSl]tji��`Ch�R[QOh�g�qSR[abR[hFi�R[XZ�@R[n?oNiW�¤R[X.k?R[h<M� ��/R[XZqSR[o�R[h¾¼FQSRc�ZQSl]t�R[QOhGR[h�© R[�¤R[QS� �

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(a, b, c) ⊂ R3 , #&-gK!%�;h.BE09iRWjOE.0,AQG-0#?, %�, -ba)F ;8.G(5F�F

R3 , #&-k1l35%76c, %�mn#&%�.)[o^#?, >&>?, #?`�'pR:9<, -0-G, -/F7#?,q.G(5F/rGsutwvyx5z|{�z7xT}5~htwv

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(a1, b1, c2)× (a2, b2, c2) = (b1c2 − b2c1, c1a2 − c2a1, a1b2 − a2b1)

��(v × w = −w × v aG#?F7Rn#&=8=_, %

v × v = 0a�(+>?F ;8#?F7R��7,O.0,OF4�!>?, =_, -hR

v 6= 0, #&-k�lE0>&>&R�, #&>?, %TS)�W,OF7m�, J<, -

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¦ -G.�aG.G(5.0.B§<aG`T(+-k§<;5E¢-G;Tm¨=E0>&R7#&60>&§©R7%7#&60>?,OFu[m�;5% (+EBYZ'0#&-IªW(+=8#&>&R�;5-k.G(+-0-�J<,T(+-hR|m�;5%7R�, R^')(+*�, -�F�;5>&>y�

�^;5R�§<, RTaB=§©F ;5-bS\«=   (+'0% 1843

an*c, #�, #&-G, =¬£h6)(+jO#?, %7Jp(+-0JD=8#&R�F�, #&-G, %k�G% (+E¨.BE0%�`�'¨�^E0*0>&#&-ban#?F�R©ªW(+=8#&>&R�;5-¨.G(+-0-¨.B#?,�!%7>?, EG` 'pR7E0-0J�J<, 9<;5=8=_, -bS�£i, #&-G,���-pR�.0,O`�9iE0-0Jq')(+Rl, %�#&-�.B#?,�­�%�;5E0JM')(+=®­�%7#?.BJ<,�, #&-0J<, 9<, %7*0RT�

i2 = j2 = k2 = ijk = −1¯±°�²T³µ´f¶©³¸·¹·¹³¸ºu»P¼f½�¾�ºu¿WÀLºu»n³¹·µÁy½A¿pÂ∗ Ã7ÄAÅAÆ Â † Ã7ÄAÇAÆ

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����� ��� ��� ��������������������������! #"%$����&�(')&�%&

ªW(+=8#&>&R�;5-k')(+R7R�,�.B#?,�*lE)(+R�, %7-0#?;5-G, -�, -hR�.0,O`�9iRT��F�R (+R7R4=8#&R4ef%7#&6c, >&-�9h>�(+6060Rl,OFl(+EBY�, #&-0=q(+>f=8#&R+*lE)(5.B%7EB�6�, >&--,

R4 = {(a, b, c, d) | a, b, c, d ∈ R}£i`�'0%�, #&*�,�-G;i`�'b�

1 := (1, 0, 0, 0), i := (0, 1, 0, 0), j := (0, 0, 1, 0), k := (0, 0, 0, 1)

�!#&-I(+>&>&J<, =_, #&-G,OF^�!>?, =_, -hR(a, b, c, d) ∈ R4 F `�'0%�, #&*�, -�mn#&%l(+>?F

q = (a, b, c, d) = a+ b i+ c j + d k

"$#&%4.0,AQG-0#?, %�, -�, #&-�K!%�;i.BE09hRW(+EBYR4 a)#&-G.0, =®mn#&%^.B#?,�K!%�;i.BE09hR�,�(+EGF

i, jE0-G.

kY±;5>&J<, -G.0, %7=q(/.<, -�YZ,OF�R¤�

>?, J<, -b�i2 = j2 = k2 = −1, i j = −j i = k, j k = −k j = i, k i = −i k = j

­�, #!HkE0>&R7#&60>&#&jO#?, %�, - N<;5-q1 = a1 + b1 i + c1 j + d1 k

=8#&Rq2 = a2 + b2 i + c2 j + d2 k

, -pR�F7R�, 'G, -16ef, %7=_,Ma)m�, >?` 'G,�F7#?` '�mn#?,O.0, %�-)(5`�'

1aiaj

E0-G.k

F�;5%7R7#?, %�, -�>�(5F�F�, -bS

q1q2 = (a1a2 − b1b2 − c1c2 − d1d2) +

(a1b2 + b1a2 + c1d2 − d1c2) · i +

(a1c2 − b1d2 + c1a2 + d1b2) · j +

(a1d2 + b1c2 − c1b2 + d1a2) · k

­�, =_, %79<, -GF7m�, %7Rl#?F7RT�

(a+ bi+ cj + dk)(a − bi− cj − dk) = a2 + b2 + c2 + d2 + 0 · i+ 0 · j + 0 · k

H�(+-¢-G, -0-pRq := a− bi− cj − dk .G(5F�jOE

q := a+ bi+ cj + dkt+x1032A~1465±sAz v|s�*lE)(+R�, %7-0#?;5-bS

|q| :=√a2 + b2 + c2 + d2

'G, #7.MR%8ls vZz:934�N<;5-qS ¦ >?F�;

q · q = |q|2.

ªl#?, % (+EGF�F , 'G, -�mn#&%nF�;+Y±;5%7RTai.G(/. �¤,O.0,OF;*lE)(+R�, %7-0#?;5-q 6= 0

, #&-G,W\«-pN<, %�F�,^*�, jOd0JM>&#?`�'¢.B#?,OF�, %nHIE0>&R7#&60>&#&95(+R7#?;5-*�,OF7#&R7jORTaB-)�+=8>&#?`�'b�

q−1 =1

|q|2 · q

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����� � � ���������� �������������� � �������������� �����

R4 =8#&R^9<;5=86�;5-G, -pR�, -pm�, #?F�, % ¦ .0.B#&R7#?;5-�E0-G.�.0, %l;5*�, -�, %79i>��+%7R�, -�HIE0>&R7#&60>&#&9M(+R7#?;5-�#?F�Rl, #&-���� � 5ys"!ut$#5z|{0sAzE0-G.Pmn#&%�.D.0, %%��� � 5±s"!ut$#5z|{0sAz�}psAz'&�~ 9Mv|sAz 0 5±x10cs 0XJ<, -)(+-0-pRTS��!%qmn#&%�.�a�ª^(+=8#&>&R�;5-�jOE0%q�!'0%�,Ma�=8#&R�.0, =­�EG`�'GF7R (+*�, -

H(+-0J<,O.0, E0R�, RTS

\«-�, #&-G, = £i`�'0#?,AYZ9<35%76c, %©J<, >&R�, -�(+>&>?,�1435%76�, % (w@i#?;5=_,g(+E .<, %¢H)(+*AS��^#?,kHIE0>&R7#&60>&#&95(+R7#?;5-�#?F7R¢(+>?F�;X-0#?`�'hR9<;5=8=�E0R (+R7#&N�S�^#?,W9<;5=860>?,A@B, -gC�(+'0>?, -

a+ bi95(+-0-¢=q(+-I(+>?F4F76c, jO#?, >&>?, *lE)(+R�, %7-0#?;5-G, -I.0, %4�G;5%7=

a+ bi+ 0 · j + 0 · k(+E�,b(5F�F�, -ba0F�;8.G(5F�F

R ⊂ C ⊂ H.

�Ld0%l(+>&>?,�.B#?,OF�,�1l35%76c, %4*0jOm�S�£i`�'0#?,AYZ9<35%76c, %l#?F7R�.B#?,�­�, R7% (+J<F¤YZE0-09hR7#?;5-k=�E0>&R7#&60>&#&9M(+R7#&Nca�.�S 'bSG,OFnJM#&>&R

|q1 · q2| = |q1| · |q2|.�Ld0%

C>&#?,AYZ, %7Rl.B#?,OF�.0, -.-�/�s 510 ~ 9<}5z:9Mv|s32�9<v54O�

(a21 + b21) · (a2

2 + b22) = (a1b1 − a2b2)2 + (a1b2 + b1a2)

2

E0-G.©YZd0%H

>&#?,AYZ, %7R�.B#?,OFl.0, - r 5ysAz60 ~ 9<}5z:9Mv|s32�9<v5487

(a21 + b21 + c21 + c21)(a

22 + b22 + c22 + d2

2) = (a1a2 − b1b2 − c1c2 − d1d2)2 +

(a1b2 + b1a2 + c1d2 − d1c2)2 +

(a1c2 − b1d2 + c1a2 + d1b2)2 +

(a1d2 + b1c2 − c1b2 + d1a2)2

�^#?,OF_9M(+-0-P=q(+-D-)(+R7d0%7>&#?` '$.BE0%�` ' ¦ EGF7=�E0>&R7#&60>&#&jO#?, %�, - , #&-BYy(5`�'�-)(5`�'0%�,O`�'0-G, -ban(+*�, %_m�, -0-$N<;5-D\«'0-G, -N<, %7>�(+-0JMR4mnd0%�.0,Ma

(12 + 22 + 32 + 42)(52 + 62 + 72 + 82)

(+>?Fn£hE0=8=_,4N<;5-4*lE)(5.B% (+R�, -©jOE¡F�` '0%�, #&*c, -bahm��+%�, -©F7#?,4;5'0-G,+*lE)(+R�, %7-0#?;5-G, -©jO#?, =8>&#?`�'¢(+EBY¥J<,OF `�'0=8#?F�F�, -

. . .Sf��F�JM#&*0R:(+EG`�'2-G;h` '/, #&-G, -:9;�<�hv=0 ~ 9<}5z:9Mv|s32�9<v54OS!�^#?,OF�, %�')�+-0JMR=8#&R�.0, -/F�;5J<, -)(+-0-hR�, -?>�t+vyx10 5±x10cs 0

;i.0, %@>�t+v 9�A+s 0¡jOEGF (+=8=_, -baG, #&-G, = *�,OF�;5-G.0, %�, -kK!%�;i.BE09hR4(+EBYR8 aim�, >?` 'G,OF^(+*�, %�-0#?`�'pRl=_, '0%4(5F�F�;5jO#�(+R7#&N

#?F7RTS0��(+-0-�#?F7R4(+*�, %l£i`�'0>&EGF FTSOSOS7;h.0, %u[

BDCFEGB HJILK�MONDPQSR=T�QUN�QVIUQSWVX.P+N�YZISQL[�N�Q

"$#&%nm�;5>&>?, -I�^%�, 'hE0-0J<, -k#&=R3 E0-hR�, %�F7EG`�'G, -bSc��(+*c, #�*�, jT, #?`�'0-G, -Imn#&%�=8#&R

R(v, ϕ) : R3 −→ R3

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����� ��� ��� ��������������������������! #"%$����&�(')&�%&

.B#?,��^%�, 'pE0-0J_E0= .0, -¢"$#&-09<, >ϕ ∈ [0, 2π)

=8#&R�.0, % ¦ ` 'GF�,vSB��(+*c, #�#?F�R

v ∈ R3 , #&-¢o�, 9iR�;5%�.0, %����+-0J<,� a |v| = 1

S� $�����"��� � F�#?F�R�-0#?` 'pR!F�`�'hm�, %�, #&-0jOEGF�, 'G, -ba5.G(5F�FL.B#?,�HI, -0J<,n(+>&>?, %��l%�, 'hE0-0J<, -8#&=

R3 aw.B#?,�mn#&%�=8#&RSO3*�, jT, #?`�'0-G, -bah*c, jOd0JM>&#?` '�.0, %�ª4#&-hR�, %�, #&-)(+-G.0, % (+EGF7d0'0%7E0-0Jq, #&-G,��W%7E0606c,n#?F�RTS<" , -0-�(+*�, % jOm�, #��^%�, 'pE0-0J<, -

R(v, ϕ)E0-G.

R(w,ψ)J<, J<, *c, -kF�#&-G.�aBm�(5F�#?F�R�.G(+-0-�#&'0%4K!%�;i.BE09hRu[�\«-��G;5%7=_, >&-b�

R(v, ϕ)R(w,ψ) = R(?, ?)

C�E0-)�5`�'GF�R�, #&-0=q(+>L9<35-0-G, -�mn#&%^Y±,OF7R�F7R�, >&>?, -bac.G(5F�FW,OFW(+EBY�.B#?,� l, #&'G, -BYZ;5>&J<,8(+-09<;5=8=R���.B#?,��^%7E0606�,SO3#?F7Rn-0#?` 'pR^(+*�, >?F�` 'b�

90°

90°

90°

90°

"$#&%�9<35-0-G, -¢�^%�, 'pE0-0J<, -¢=8#&RnH�(+R7%7#&jT, -�*�,OF�` '0%�, #&*c, -ba0(+*�, %�=8#&R;*lE)(+R�, %7-0#?;5-G, -¢J<, 'pR�,OF�Nh#?, >�, >?, Jp(+-pR�, %T�"$#&%q*�, jT, #?`�'0-G, - .B#?,IH�, -0J<,I(+>&>?, %#*lE)(+R�, %7-0#?;5-G, -�.0, %¡�);5%7=

u1 · i + u2 · j + u3 · k=8#&R

H0S�"$#&%

#?.0, -pR7# QGjO#?, %�, -�, #&-G, -�o�, 9iR�;5%u = (u1, u2, u3) ∈ R3

=8#&R4.0, = *lE)(+R�, %7-0#?;5-u = 0 + u1 · i+ u2 · j + u3 · k ∈ H0.

HI#&RnE0-GF�, %�, =®�^%�, 'pmn#&-09<, >ϕ

E0-G.k.0, %4�^%�, ')(5`�'GF�,v

.0,AQG-0#?, %�, -kmn#&%�, #&-km�, #&R�, %�,OF+*lE)(+R�, %7-0#?;5-

q := cos(ϕ) + sin(ϕ) · v

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E0-G.�, #&-G, ¦ *0*0#&>?.BE0-0JDq : R3 −→ R3, w 7→ q · w · q.

¦ EBY�.0, %�%�,O` 'pR�, -I£i, #&R�,WF7R�, 'hR�, #&J<, -hR7>&#?`�'�mn#?,O.0, %n, #&- *lE)(+R�, %7-0#?;5- �B%�,O`�'0-G, -k£h#?,^-)(5`�'baB.G(5F�F�,OF�mn#?,O.0, %�#&-H0

>&#?, JMR�E0-G.�F ;5=8#&R�=8#&R�, #&-G, =�o�, 9iR�;5%l(+EGFR3 #?.0, -hR7# QGjO#?, %7R4m�, %�.0, -I95(+-0-bS

� ��� � ����� ��������� ���Dq = R(v, 2ϕ) �

� � ����Z "���"� ����� �

• i(x · i+ y · j + z · k)i = x · i − y · j − z · k S ¦ >?F�;q, -hR�F760%7#?`�'hRDi

, #&-G, %180◦

�|�^%�, 'hE0-0J�E0= .B#?,x� ¦ `�'GF�,MS

• q1 = 1√2

+ 1√2· i Sh"$#&% ')(+*�, -

q21 = i

ai(+>?F ;, -hR�F760%7#?`�'hRDq1

, #&-G, %90◦

�|�^%�, 'pE0-0J8E0= .B#?,x� ¦ `�'GF�,MS

• �W, -)(+EGF ;_, -pR�F760%7#?` 'pRDq2

=8#&Rq2 = 1√

2+ 1√

2· j , #&-G, %

90◦�|�^%�, 'pE0-0J¡E0=®.B#?,

y� ¦ `�'GF�,MS

• q1q2 = 12 + 1

2 · i+ 12 · j + 1

2 · kS ¦ >?F�;8, -hR�F760%7#?`�'hR

Dq1q2

, #&-G, %120◦

�|�^%�, 'pE0-0J¡E0=(12 ,

12 ,

12)

S

• q2q1 = 12 + 1

2 · i+ 12 · j − 1

2 · kS ¦ >?F�;8, -hR�F760%7#?`�'hR

Dq1q2

, #&-G, %120◦

�|�^%�, 'pE0-0J¡E0=(12 ,

12 ,−1

2 )S

" , -0-�mn#&%L, #&-:KL(+606GF `�'0#&>?.:*)(5F7R�, >&-ba+,OFf=8#&R!£i`�'0-hd0%�, -q(+-*�, >&#?, *0#&JlNi#?, >?, -:K!E0-09hR�, -�#&= l(+E0=¨*�,AY±,OF7R7#&J<, -� ;i.0, %�Y±,OF7R7')(+>&R�, -�>�(5F�F�, - * E0-G.©,OF�.G(+-0-�-)(5` 'G, #&-)(+-G.0, %��¤, m�, #&>?F�E0=���(����W% (5.q#&-¢.B#?,^.B%�, #�1l;h;5%�.B#&-)(+R�, -B�%7#?`�'hR7E0-0J<, -�.B%�, 'G, -bSOSOS

SOSOS7.G(+-0-IF�#&-G.©jOE0= £i`�'0>&EGF�F^(+>&>?,�£i`�'0-hd0%�,�mn#?,O.0, %4, -hR|mn#&%7%7R3,c�W, % �^%7E0-G.��

i4 = j4 = k4 = 1.