494 Homeworks (Andrew Snowden).pdf

8/19/2019 494 Homeworks (Andrew Snowden).pdf

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Math 494 • Homework 1 • Due January 12

Homework is due at the beginning of class. Late homework is not accepted. I encourage you to work

with others on homework problems, but you must write up your own solutions. Solutions must bepresented clearly, or will be marked down.

Problem 1 (4 points) [10.1.1]. Prove the following identities in an arbitrary ring R: (a) 0a = 0;(b) −a = (−1)a; (c) (−a)b = −(ab). Here −a means the additive inverse of a.

Problem 2 (6 points) [10.1.9]. In each case, decide whether the given structure forms a ring. If it is not a ring, determine which of the ring axioms hold and which fail.

(a) U is an arbitrary set, and R is the set of subsets of U . Addition and multiplication of elementsof R are defined by the rules A + B = A ∪B and A · B = A ∩B.

(b) U is an arbitrary set, and R is the set of subsets of U . Addition and multiplication of elementsof R are defined by the rules A + B = (A ∪B) \ (A ∩B) and A · B = A ∩B.

(c) R is the set of continuous functions R → R. Addition and multiplication are defined by therules (f + g)(x) = f (x) + g(x) and (fg)(x) = f (g(x)).

Problem 3 (6 points) [10.1.11]. Describe the units in each ring: (a) Z/12Z; (b) Z/7Z; (c) Z/8Z;(d) Z/nZ.

Problem 4 (4 points) [10.1.12]. Prove that the units of the Gaussian integers are {±1,±i}.

Problem 5 (4 points) [10.1.13]. An element x of a ring R is called nilpotent if some power of xis zero. Prove that if x is nilpotent then 1 + x is a unit.

Problem 6 (6 points) [10.3.3]. For which integers n does x2 +x+ 1 divide x4 + 3x3 +x2 + 6x+ 10in Z/nZ[x]?

Problem 7 (4 points) [10.3.4]. Prove that in the ring Z[x], (2) ∩ (x) = (2x).

Problem 8 (6 points) [10.3.19]. Let p be a prime number, and let R be a commutative ring inwhich p = 0. Show that the map R → R defined by x 7→ x p is a ring homomorphism.


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Math 494 • Homework 2 • Due January 26



Problem 1 (4 points) [10.3.8]. Describe the kernel of the following ring homomorphisms:

(a) R[x, y] → R defined by f 7→ f (0, 0).

(b) R[x] → C defined by f 7→ f (2 + i).

Problem 2 (6 points) [10.3.24]. (a) The nilradical N of a ring R is the set of its nilpotentelements. Prove that N is an ideal. (b) Determine the nilradicals of Z/(12), Z/(n), and Z.

Problem 3 (2 points) [10.1.14]. Prove that the product set R × R0 of two rings is a ring with

component-wise addition and multiplication:

(a, a0) + (b, b0) = (a + b, a0 + b0), (a, a0)(b, b0) = (ab,a0b0)

This ring is called the product ring .

Problem 4 (6 points) [10.3.34]. An element e of a ring S is called idempotent if e2 = e. Notethat in a product R ×R0 of rings, the element e = (1, 0) is idempotent. The object of this exerciseis to prove a converse.

(a) Prove that if e is idempotent then e0 = 1 − e is also idempotent.

(b) Let e be an idempotent of a ring S . Prove that the principal ideal eS is a ring, with identityelement e. [Note: it is typically not a subring of S , since it won’t contain 1 unless e = 1.]

(c) Let e be an idempotent, and let e0 = 1 − e. Prove that S is isomorphic to the product ring(eS ) × (e0S ).

Problem 5 (6 points) [10.4.7]. Let I and J be ideals of a ring R such that I + J = R.

(a) Prove that I J = I ∩ J .

(b) Prove the Chinese remainder theorem : for any pair of elements a, b ∈ R there is an elementx ∈ R such that x = a (mod I ) and x = b (mod J ). (The notation x = a (mod I ) meansx− a ∈ I .)

Problem 6 (4 points) [10.4.8]. Let I and J be ideals of a ring R such that I +J = R and IJ = 0.

(a) Prove that R is isomorphic to the product (R/I ) × (R/J ).


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(b) Describe the idempotents corresponding to this product decomposition (see the previous ex-ercise).

Problem 7 (4 points) [10.5.15]. Let a be an element of a ring R, and let R0 be the ring obtainedfrom R by adjoining an inverse of a. Prove that R0 is the zero ring if and only if a is nilpotent.

Problem 8 (4 points) [10.6.2]. Prove that an integral domain with finitely many elements is afield.

Problem 9 (2 points) [10.6.5]. Is there an integral domain containing exactly 10 elements?

Problem 10 (4 points) [10.6.3]. Let R be an integral domain. Prove that the polynomial ringR[x] is an integral domain.

Problem 11 (2 points) [10.7.1]. Prove that the maximal ideals of the ring of integers are theprincipal ideals generated by prime integers.

Problem 12 (4 points) [10.7.7]. Prove that the ring F2[x]/(x3 + x + 1) is a field, but that

F3[x]/(x3 + x + 1) is not a field.

Problem 13 (4 points) [10.7.10]. Let R be a ring, with M an ideal of R. Suppose that everyelement of R that is not in M is a unit of R. Prove that M is a maximal ideal and that moreoverit is the only maximal ideal of R.

Problem 14 (6 points) [10.misc.21]. Let f , g be polynomials in C[x, y] with no common factor.Prove that the ring R = C[x, y]/(f, g) is a finite dimensional vector space over C.

Problem 15 (6 points) [10.misc.23]. Let f (x), g(x) be polynomials with coefficients in a ring Rwith f 6= 0. Prove that if the product f (x)g(x) is zero, then there is a non-zero element c ∈ R suchthat cg(x) = 0.


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Math 494 • Homework 3 • Due February 2



Problem 1 (2 points) [11.2.1b]. Prove that Z[x] is not a PID.

Problem 2 (2 points) [11.5.2]. Factor 30 into primes Z[i].

Problem 3 (2 points) [11.6.3]. Let d and d0 be distinct square-free integers. Prove that Q(√ d)

and Q(√ d0) are diff erent subfields of C.

Problem 4 (2 points) [11.7.1]. Prove that 2 and 3 and 1 +√ −5 and 1 − √ −5 are irreducible

elements of the ring Z[√ −5].

Problem 5 (2 points). An ideal I of a commutative ring R is called prime if the following conditionholds: given elements a, b ∈ R such that ab ∈ I , either a ∈ I or b ∈ I . Show that I is prime if andonly if R/I is a domain. In particular, maximal ideals are prime.

Problem 6 (2 points). Let R be an integral domain and let π be a non-zero non-unit of R. Showthat π is prime if and only if the principal ideal (π) is prime.

Problem 7 (12 points). Let ζ = e2πi/3 and R = Z[ζ ]. (This is sometimes called the ring of Eisenstein integers.) Let p 6= 3 be a prime integer.

(a) Show that R is a Euclidean domain, and thus a PID, and thus a UFD.

(b) Show that either p is prime in R or p = ππ for some prime π of R.

(c) Show that the following conditions are equivalent: (i) p factors as ππ in R; (ii) p can be writtenin the form a2 + ab + b2 with a, b ∈ Z; (iii) the polynomial x2 + x + 1 has a root in F p; (iv) p = 1 (mod 3).

(d) Factor 30 into primes in R.

The ideas we used in analyzing Z[i] can be used to do this problem.

Problem 8 (16 points) Let R be a UFD with field of fractions F . A polynomial f (x) = anxn +

· · · + a0 in R[x] is primitive if gcd(a0, . . . , an) = 1, i.e., no element of R properly divides all the ai.

(a) Let f (x) ∈ F [x] be non-zero. Show that there is an element c of F ×, unique up to multiplicationby units of R, such that c−1f (x) belongs to R[x] and is primitive. The element c is called the

content of f , and denoted c(f ).

(b) Let f (x) ∈ F [x]. Show that f (x) ∈ R[x] if and only if c(f ) ∈ R, and f (x) is primitive if andonly if c(f ) is a unit of R.


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(c) Let f (x), g(x) ∈ F [x] be non-zero. Show that c(fg) = c(f )c(g). [Hint: reduce to the casec(f ) = c(g) = 1. If c(fg) 6= 1, choose a prime π of R dividing c(fg), and reduced mod π. Usethe fact that (R/π)[x] is a domain to obtain a contradiction.]

(d) Show that an irreducible element f (x) of R[x] is prime. [Hints: (i) Show that f (x) is primitive.(ii) Show that f (x) is irreducible in F [x], and thus prime in F [x]. (iii) Supposing f (x) dividesg(x)h(x) in R[x], conclude f (x) divides either g(x) or h(x) in F [x]; (iv) Finally, show thatf (x) divides either g(x) or h(x) in R[x].]

(e) Show that R[x] is a UFD.

The section of the textbook on Gauss’s Lemma (or the Wikipedia article) could be helpful.


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Problem 1 (6 points) [11.7.3]. Let R = Z[√ −5]. Determine whether or not the subgroup of R

generated by the given elements is an ideal:

(a) 5 and 1 +√ −5

(b) 7 and 1 +√ −5

(c) 4− 2√ −5 and 2 + 2√ −5 and 6 + 4√ −5

Problem 2 (4 points) [11.7.4]. Let R be the ring of integers in an imaginary quadratic field, and

let A be a non-zero ideal of R. Prove that there exists an integer α ∈ Z and an element β ∈ R suchthat A is generated as a subgroup of R by α and β . (That is, A = {nα + mβ | n, m ∈ Z}.)

Problem 3 (4 points) [11.8.1]. Let R = Z[√ −6]. Factor (6) into primes ideals explicitly.

Problem 4 (4 points) [11.8.2]. Let R = Z[√ −3]. This is not the ring of integers in Q(√ −3).

Let A be the ideal (2, 1 +√ −3). Show that AA is not a principal ideal, and so the Key Lemma is

not true for R.

Problem 5 (4 points) [11.8.3]. Let R = Z[√ −5]. Determine whether or not 11 is an irreducible

element of R and whether or not (11) is a prime ideal of R.

Problem 6 (4 points) [11.8.6]. Let R = Z[√

−5]. Factor (14) into prime ideals explicitly.

Problem 7 (4 points). Let R be the ring of integers in an imaginary quadratic field and let Aand B be non-zero ideals of R. We say that A and B are coprime if there exists no non-zero primeideal dividing both of them.

(a) Show that A and B are coprime if and only if A + B = (1).

(b) Suppose A and B are coprime. Show that R/AB is isomorphic to R/A× R/B. [Hint: HW2may be useful!]

Problem 8 (4 points). Let R be the ring of integers in an imaginary quadratic field and let P bea non-zero prime ideal of R.

(a) Show that there exists an element α ∈ R such that P divides (α) but P 2 does not divide (α).(b) Show that every ideal of R/P n is of the form (αi) for some 0 ≤ i ≤ n.


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Problem 9 (6 points). Let R be the ring of integers in an imaginary quadratic field and let A bea non-zero ideal of R.

(a) Suppose that S and T are rings in which every ideal is principal. Show that the same is truefor S × T .

(b) Show that every ideal of R/A is principal. [Hint: factor A into prime ideals and use problems 7and 8.]

(c) Let x ∈ A be a non-zero element. Show that there exists y ∈ A so that A = (x, y). [Hint:apply (b) to the ring R/(x).]

In particular, every ideal of R is generated by at most two elements. This is true whenever R is thering of integers in a number field (not just an imaginary quadratic field), by the same proof.


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Problem 1 (2 points) [12.1.1]. Let R be a ring, considered as an R-module. Determine allmodule homomorphisms R → R.

Problem 2 (4 points) [12.1.6]. A module is called simple if it is not the zero module and if ithas no proper submodule.

(a) Prove that any simple module is isomorphic to R/M , where M is a maximal ideal.

(b) Prove Schur’s lemma: if φ : S → S 0 is a homomorphism of simple modules then φ is either zeroor an isomorphism.

Problem 3 (2 points) [12.2.1]. Let R = C[x, y] and let M be the ideal (x, y). Prove or disprove:M is a free R-module.

Problem 4 (2 points) [12.2.3]. Let I be an ideal of a ring R. Prove or disprove: If R/I is a freeR-module then I = 0.

Problem 5 (4 points) [12.2.4]. Let R be a ring and let V be a free R-module of finite rank.Prove or disprove:

(a) Every set of generators contains a basis.

(b) Every linearly independent set can be extended to a basis.

Problem 6 (2 points) [12.5.2]. Find a ring R and an ideal I of R that is not finitely generated.

Problem 7 (4 points) [12.4.10]. Let φ : Zk → Zk be a homomorphism given by multiplicationby an integer matrix A. Show that the image of φ has finite index if and only if A is nonsingularand that if so, then the index is equal to | det A|.

Problem 8 (2 points) [12.6.4]. Determine the number of isomorphism classes of abelian groupsof order 400.

Problem 9 (2 points) [12.misc.9]. Prove that the multiplicative group Q× of rational numbersis isomorphic to the direct sum of a cyclic group of order 2 and a free abelian group with countably

many generators.

Problem 10 (4 points). Let R be a noetherian ring. Show that there exists an integer n > 0,depending only on R, such that if x ∈ R is nilpotent then xn = 0.


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Problem 11 (4 points). Let R be a ring and let I be an ideal of R. Suppose that R/I is noetherianand that every ideal of R contained in I is finitely generated. Prove that R is noetherian.

Problem 12 (6 points). Let α be a positive real number. Let R ⊂ C[x, y] be the C-span of all

monomials of the form xi

yj

with j ≤ αi.

(a) Show that R is a subring of C[x, y].

(b) Assume α is rational. Show that R is a quotient of C[z1, . . . , zn] for some n, and thereforenoetherian.

(c) Assume α is irrational. Show that R is not noetherian.

This example shows that a subring of a noetherian ring need not be noetherian.


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Math 494 • Homework 6 • Due March 16

Homework is due at the beginning of class. Late homework is not accepted. I encourage you to workwith others on homework problems, but you must write up your own solutions. Solutions must bepresented clearly, or will be marked down.

Problem 1 (4 points) [13.1.3]. Let R be an integral domain containing a field F as a subringand which is finite dimensional when viewed as an F -vector space. Prove that R is a field.

Problem 2 (2 points). Give a counterexample to Problem 1 when R is not assumed to be anintegral domain. (That is, give a commutative ring R containing a field F such that R is finitedimensional over F but R is not a field.)

Problem 2 (4 points) [13.2.1]. Let α be the real cube root of 2. Compute the minimal polynomialof 1 + α2 over Q.

Problem 3 (4 points) [13.2.6]. Let β = ζ 3

√ 2, where ζ = e2πi/3, and let K = Q(β ). Prove that−1 cannot be written as a sum of squares in K .

Problem 4 (4 points) [13.3.1]. Let K/F be a field extension, and let α be an element of K suchthat F (α) has degree 5 over F . Prove that F (α2) = F (α).

Problem 5 (4 points) [13.3.2]. Let ζ = e2πi/7 and let η = e2πi/5. Prove that η 6∈ Q(ζ ).

Problem 6 (4 points) [13.3.8]. Let K/F be a field extension such that K = F (α, β ) where αand β have relatively prime degrees n and m over F . Prove that [K : F ] = nm.

Problem 7 (4 points) [13.3.10]. Let α and β be complex numbers. Prove that if α + β and αβ are algebraic then α and β are algebraic.

Problem 8 (14 points). Let F be a field. A polynomial f ∈ F [t] is separable if it has no repeatedroots. This means that if K/F is any field extension then, in the ring K [t], the element f is notdivisible by (t− a)2 for any a ∈ K . An element of an extension K/F is separable if it is algebraicand its minimal polynomial is separable.

(a) Let f ∈ F [t] be an irreducible polynomial that is not separable. Prove that f 0 = 0, where f 0denotes the derivative of f . [Hint: pick K/F and a ∈ K such that (t− a)2 divides f . Then f is the minimal polynomial of a but f 0(a) = 0 too.]

(b) Suppose that F has characteristic 0. Show that any irreducible polynomial in F [t] is separable.

Thus separability is not an interesting property in characteristic 0. For the remainder of this problem,we assume F has characteristic p. Note that f 0 = 0 does not imply f is a constant in characteristic p,e.g., f (t) = t p has f 0 = 0.

(c) Let f ∈ F [t] be irreducible. Show that there exists an irreducible separable polynomial g ∈ F [t]such that f (t) = g(t p

n

) for some n.


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(d) The field F is called perfect if every element is a pth power (e.g., F p). Show that if F isperfect then any irreducible polynomial in F [t] is separable. [Hint: show that if g ∈ F [t] is anypolynomial then g(t p) = h(t) p for some other polynomial h ∈ F [t], and apply (c).]

(e) Let K/F be an extension, and let a ∈ K be algebraic over F . Show that a pn

is separable overF , for some n > 0.

(f) Let F = F p(u). Give an example of an extension K/F and an element a ∈ K that is algebraicbut not separable over F .


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Math 494 • Homework 7 • Due March 30



Problem 1 (10 points). Let L be a field and let K be a subset of L. Then K is a subfield of L if and only if it satisfies the following conditions:

(i) 1 ∈ K .(ii) If x, y ∈ K then x + y ∈ K .

(iii) If x ∈ K then −x ∈ K .(iv) If x, y ∈ K then xy ∈ K .(v) If x

∈K and x

6= 0 then 1/x

∈K .

In this problem, you will examine the logical interdependence of these conditions.

(a) Show that conditions (i), (ii), (iii), and (v) are essential by way of example. That is, for eachof these conditions, construct a subset K of a field L that is not a subfield, but that satisfiesall the other conditions.

(b) Show that condition (iv) is implied by the other four conditions if the characteristic of L isnot 2.

(c) Let L = F2(x, y). Let K be the smallest subset of L containing x and y and satisfyingconditions (i), (ii), (iii), and (v). Show that if f ∈ K then ∂ x(∂ y(f )) = 0, where ∂ x and ∂ yare the partial derivatives with respect to x and y . In particular, K does not contain xy , andthus does not satisfy (iv).

Hint: K be obtained inductively as follows. Let K 0 be the F2-span of 1, x, and y in L. Havingdefined K n, let K n+1 be the subspace of L spanned over F2 by K n and the elements 1/f withf ∈ K n non-zero. Then K is the union of the K n. You can use this to inductively provestatements about elements of K .

(d) Assuming L has characteristic p > 0, show that condition (iii) is implied by the other condi-tions.

Problem 2 (4 points). Neither 3 nor 5 is a square in the field F7, and so F7(√

3) and F7(√

5)are degree 2 extensions of F7. They are therefore isomorphic, by the classification of finite fields.Explicitly construct an isomorphism between them.

Problem 3 (2 points). Let K be a field of characteristic p, and let ζ ∈ K be a pnth root of 1, forsome n ≥ 0. Show that ζ = 1.


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Problem 4 (2 points). Let q be a power of an odd prime p, and let x ∈ F×q . Show that

x(q−1)/2 ∈ {±1} and that x(q−1)/2 = 1 if and only if x is a square.

Problem 5 (4 points). Let q be a power of an odd prime p. Show that every element of Fq is a

sum of two squares.

Problem 6 (2 points). Let q be an even power of a prime p. Show that −1 is a square in Fq.

Problem 7 (4 points). Let q be a prime power. How many elements of Fq are cubes?

Problem 8 (4 points). Let F be an algebraically closed field and let f ∈ F [x1, . . . , xn] be a non-constant polynomial. Show that there exists a point (a1, . . . , an) ∈ F

n such that f (a1, . . . , an) = 0.

Problem 9 (8 points). A Laurent series is a formal series of the form

a−nt−n + · · · + a−1t

−1 + a0 + a1t + a2t2 + · · ·

where the ai are complex numbers and n is an integer. In words, a Laurent series is a powerseries where the powers of t are allowed to be both positive and negative, but only finitely manynegative powers are allowed. The set of Laurent series is denoted C((t)). One defines addition andmultiplication of Laurent series in the obvious manner, and this makes C((t)) into a ring.

(a) Show that C((t)) is in fact a field.

Let C((t1/n)) be the field of Laurent series in t1/n. This contains C((t)), and so can be thought of asan extension field of C((t)).

(b) Show that C((t1/n)) has degree n over C((t)).

(c) Show that every element f ∈ C((t)) has the form gn for some g ∈ C((t1/n)).

(d) Show that C((t1/2

)) is the unique degree 2 extension of C((t)).In fact, the union of the C((t1/n)) is the algebraic closure of C((t)).


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Math 494 • Homework 8 • Due April 6



Problem 1 (6 points) [14.1.6]. Determine the degrees of the splitting fields of the followingpolynomials (over Q): (a) x4 − 1; (b) x3 − 2; (c) x4 + 1.

Problem 2 (4 points) [14.1.8]. Let ζ = e2πi/5. Prove that Q(ζ ) is the splitting field of x5 − 1over Q and determine its Galois group.

Problem 3 (4 points) [14.1.15]. Suppose that K/F is a Galois extension with Galois group theKlein 4-group (i.e., Z/2Z × Z/2Z). Show that K/F is biquadratic, i.e., K = F (√ a,

√ b) for some

a, b ∈ F .

Problem 4 (6 points) [14.1.15]. Let K/F be a Galois extension with Gal(K/F ) ∼= Z/2Z×Z/12Z.How many intermediate fields L are there such that (a) [L : F ] = 4; (b) [L : F ] = 9; (c) Gal(K/L) ∼=Z/4Z?

Problem 5 (6 points) [14.4.2]. Let α = 3√

2 and ζ = e2πi/3 and β = αζ .

(a) Prove that for all c ∈ Q the element α + cβ is a root of a sixth degree polynomial of the formx6 + ax3 + b with a, b ∈ Q.

(b) Prove that the minimal polynomial for α + β is cubic.

(c) Prove that the minimal polynomial for α− β has degree 6.

Problem 6 (10 points) [14.4.3]. For each of the following sets of automorphisms of the fieldC(y), determine the group of automorphisms they generate and find the fixed field explicitly: (a)σ(y) = y−1; (b) σ(y) = iy; (c) σ(y) = −y and τ (y) = y−1; (d) σ(y) = ζ y and τ (y) = y−1, whereζ = e2πi/3; (e) σ(y) = iy and τ (y) = y−1.

Problem 7 (6 points) [14.4.4]. Define automorphisms σ and τ of the field C(y) as follows:

σ(y) = y + i

y − i , τ (y) = i(y − 1)y + 1

.

Show that the subgroup G of Aut(C(y)) they generate is isomorphic to A4. Then determine thefixed field C(y)G.

Problem 8 (4 points) [14.misc.4]. Let K/F be a Galois extension with group G, and let H bea subgroup of G. Prove that there exists β ∈ K whose stabilizer in G is exactly H .


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Homework is due at the beginning of class. Late homework is not accepted. I encourage you to workwith others on homework problems, but you must write up your own solutions. Solutions must bepresented clearly, or will be marked down.

Problem 1 (4 points) [14.2.1]. Prove that the discriminant of a real cubic polynomial is positiveif all the roots are real, and negative if not.

Problem 2 (2 points). Let f be an irreducible cubic polynomial with rational coefficients withexactly one real root. Show that the splitting field of f (over Q) has Galois group S 3.

Problem 3 (4 points) [14.2.3]. Let f be an irreducible cubic polynomial over F , and let δ be thesquare root of the discriminant of f . Prove that f remains irreducible over F (δ ).

Problem 4 (6 points) [14.2.5]. Let f = x3 + px + q be an irreducible cubic polynomial over F ,

let K be its splitting field, let α1, α2, and α3 be the three roots of f in K , and let δ be the squareroot of the discriminant. Then K = F (α1, δ ). Give explicit formulas for α2 and α3 in terms of α1and δ (and p and q , and whatever other elements of F you’d like).

Problem 5 (4 points). Let E/K be a finite Galois extension and let L1 and L2 be intermediatefields that are Galois over K and satisfy L1 ∩ L2 = K . Show that the compositum L1L2 ⊂ E isGalois over K and that the natural map

Gal(L1L2/K ) → Gal(L1/K ) ×Gal(L2/K )

is an isomorphism.

Problem 6 (4 points). Give a Galois extension of Q that has Galois group Z/6Z. [Hint: Z/6Z is

the product of Z/2Z and Z/3Z.]

Problem 7 (4 points) [14.5.3]. Let G be a finite group. Show that there exists some Galoisextension K/F with Galois group isomorphic to G.

Problem 8 (4 points). Give an example of fields F ⊂ K ⊂ L such that L/K and K/F are Galoisbut L/F is not Galois.

Problem 9 (8 points) [14.3.4]. For each of the following polynomials, determine if it is a sym-metric function, and if so, express it in terms of the elementary symmetric functions.

(a) u21u2 + u

2

2u1 with n = 2.

(b) u21u2 + u

2

2u3 + u

2

3u1 with n = 3.

(c) (u1 + u2)(u1 + u3)(u2 + u3) with n = 3.

(d) u31 + u3

2 + · · · + u3

n.


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Problem 1 (6 points) [14.6.10]. Find a quartic polynomial over Q whose Galois group is (a) S 4,(b) D4, (c) Z/4Z.

Problem 2 (6 points) [14.6.12]. Determine the Galois groups of the following polynomials overQ: (a) x4 + 4x2 + 2, (b) x4 + 2x2 + 4, (c) x4 − 2.

Problem 3 (10 points) [14.6.14]. Let f be an irreducible quartic polynomial over F of the formx4 + rx + s, and let α1, α2, α3, α4 be the roots of f in a splitting field K . Let η = α1α2.

(a) Prove that η is the root of a sextic polynomial h(x) with coefficients in F .

(b) Assume that the six products αiαj are distinct. Prove that h(x) is irreducible, or else it hasan irreducible quadratic factor.

(c) Describe the possibilities for Gal(K/F ) in the following three cases: h is irreducible, h is theproduct of an irreducible quadratic and an irreducible quartic, and h is the product of threeirreducible quadratics.

(d) Describe the situation where some of the products αiαj are equal.

Problem 4 (2 points) [14.6.17]. Let f (x) be a quartic polynomial. Prove that the discriminantsof f and of its cubic resolvent are equal.

Problem 5 (10 points) [14.7.2]. Let a be a non-zero element of a field F and let p be a primenumber. Suppose that f (x) = x p − a is reducible in F [x]. Prove f (x) has a root in F , as follows.

(a) Show that the splitting field of f is F (ζ ), where ζ is a primitive pth root of unity. [Hint: letK be the splitting field. First show that K contains F (ζ ). Next show that the orbit of a rootof f under Gal(K/F (ζ )) must have either 1 or p elements. Since f is reducible, it must haveone element. Use this to conclude K = F (ζ ).]

(b) Let G be a group and let i : G → (Z/pZ)× be an injective homomorphism. Suppose thatf : G → Z/pZ is a function satisfying the identity f (στ ) = i(σ)f (τ ) + f (σ). Show that thereis some c ∈ Z/pZ such that f (σ) = (i(σ) − 1)c for all σ ∈ G.

Now let G be the Galois group of F (ζ )/F . Recall that there is a natural injective homomorphismi : G → (Z/pZ)× characterized by σ(ζ ) = ζ i(σ) for σ ∈ G.

(c) Let u be a root of f (x) in F (ζ ). Define a function f : G → Z/pZ by σ(u) = ζ f (σ)u. Show thatf (στ ) = i(σ)f (τ ) + f (σ), and therefore f (σ) = (i(σ) − 1)c for some c ∈ Z/pZ.

(d) Finally, show that ζ −cu is a root of f (x) in F .


17/17

Problem 6 (10 points). The goal of this problem is to show that a quadratic extension F (√ d)

sits inside a Z/4Z extension if and only if d is a sum of two squares. For example, Q(√ −1) does

not sit inside a Z/4Z extension of Q, but Q(√

2) does.

(a) Let F be a field that is not of characteristic 2. Let G be the set of all non-zero elements a ∈ F

such that a = b2 + c2 for some b, c ∈ F . Show that G is a subgroup of F ×.(b) Let K/F be a Galois extension with group Z/4Z. Let F (

√ d) be the unique intermediate field

of degree 2 over F , where d is some element of F . Show that d is a sum of two squares in F .

Hint: K = F (p a + b

√ d) for some a, b ∈ F . The generator of the Galois group carriesp

a + b√ d to

p a− b

√ d, and so their product

√ a2 − db2 is in K . Thus F (

√ a2 − db2) is an

intermediate field of K of degree 2 over F . Conclude a2 − db2 = c2d for some c ∈ F , and usethis to express d as a sum of two squares. If you use any of the assertions in this hint, be sureto prove them in your solution.

(c) Let d be an element of F that is not a square, but is a sum of two squares. Show that there existsa Galois extension K/F with group Z/4Z containing the quadratic extension F (

√ d). [Hint:

Find a, b, c ∈ F with c 6= 0 such that a2 − db2 = c2d, and use the extension F (p a + b√ d).](d) Suppose that

√ −1 ∈ F . Prove directly (without using the other parts of this problem) thatevery element of F is a sum of two squares, and that every quadratic extension of F sits insideof a Z/4Z extension.

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