v'2 v4. - UCLA Department of MathematicsD2. Describe, up to similarity, all 4 x 4 matrices A over Q...

Spring 2000

Algebra Qualifying Exam

Everyone must do two problems in each of the four sections. To pass at the Ph.D. level, you must attempt at least three 20 point prob

lems. On multiple part problems, do as many parts as you can; however, not all parts count equally.

A. GROUPS

Al.(10 points) State a theorem which classifies (i.e. lists) all finite abelian groups up to isomorphism. This means that each finite abelian group should be isomorphic to exactly one group of your list. Use your classification to list abelian groups of order 24.

A2. (15 points) Let S5 be the symmetric group on 5 letters. For each positive integer n, list the number of elements of S5 of order n. Justify your answer.

A3. (20 points) Let IF4 be the field with 4 elements. Let G = SL(2, IF4 ) be the group of 2 by 2 invertible matrices with entries in IF 4 . What is the order of G ? Show, by analysing the action of G on the lines containing the origin in (IF 4)2, that G is a simple group. (Hint: how many lines containing the origin are there?)

B. RINGS

Bl.(10 points) List, up to isomorphism, all commutative rings with 4 elements. Prove your answer.

B2.(15 points) Let p be a prime number. Show that a free Z module of rank 2 has p+ 1 submodules of index p.

B3.(20 points) Let R be a commutative noetherian ring in which each ideal I is principal and satisfies 12 = I. Show that R is isomorphic to a finite product of fields.

C. FIELDS

Cl. Let a= 1 + 3v'2 + 3v4. (a) Find the degree of a over Q. Justify your answer.

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(b) Find a normal closure of Q(a)/Q. Justify your answer.

C2. Let q be a power of a prime integer, n E N. Let k be the least positive integer such that qk - 1 ( mod n). Prove that the finite field IF qk is a splitting field of the polynomial xn - 1 over IF q·

C3. Find a subfield Fin the field of rational functions C(X) such that C(X) / F is a Galois extension with the Galois group isomorphic to the symmetric group S 3 . (Hint: Consider automorphisms of C(X) given by X f---+ ~{!!, a, b, c, d E

Z.)

D. LINEAR ALGEBRA

D 1. Let A be a linear operator on a vector space of dimension n such that Am is the zero operator for some m. (a) Prove that all eigenvalues of A are equal to zero. (b) Prove that An= 0.

D2. Describe, up to similarity, all 4 x 4 matrices A over Q such that A5 = -A 3 but A 3

-/:- 0. Justify your answer.

D3. Let End(V) be the ring of all linear operators on a finite dimensional vector space V (with respect to the addition and composition of operators). For an operator A E End(V) let LA be the subspace in End(V) generated by the powers Ai, i ~ 0. (a) Show that LA is a subring in End(V). (b) Prove that if LA is a field then the characteristic polynomial of A is a power of an irreducible polynomial.


Fall 2000

Everyone must do two problems in each of the four sections. To pass at the PhD level, you must attempt at least three 20-point problems. On multiple part problems, do as many parts as you can; however, not all parts count equally.

Groups

Al. (10 points) Let D2n be the dihedral group of order 2n with n > l. Determine the number of subgroups of D 2n of index 2, and justify your answer.

A2. (15 points) A group of order a power of a prime pis called a p-group.

A3.

Let G be a finite group. Prove that for any given prime p, there exists a unique normal subgroup N of G such that (i) G/N is a p-group and (ii) any homomorphism 7f of G into a p-group is trivial on N (that is, 1r(N) = 1).

(20 points) Let G be a finite group of order n. Suppose that G has a unique subgroup of order d for each positive divisor d of n. Prove that G is a cyclic group.

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Rings

Bl. (10 points) Let M be a module over a commutative ring A. If every increasing (resp. decreasing) sequence of A-submodules of M terminates after finite steps, the A-module M is called noetherian (resp. artinian).

(a) Prove that the Z-module Z is noetherian and non-artinian.

(b) Prove that the Z-module LJ:=1 (p-nZ/Z) is artinian and nonnoetherian.

B2. (15 points) Let A be a commutative ring with identity. Suppose that a EA is not nilpotent (that is, an -IO for all n > 0).

(a) Prove that there exists a prime ideal PC A such that a r:f. P;

(b) Give an example of a ring A and a non-nilpotent a E A such that a is contained in M for all maximal ideals NI C A. Justify your example.

B3. (20 points) Let A be a commutative noetherian ring with identity 1 -I 0. Write X (Cl) for the set of prime ideals of A containing a given ideal Cl.

Suppose that X(0) = X(a) UX(b) and X(a) nX(b) = 0 for two ideals Cl and b. Prove the following facts:

(a) A= a+ b;

(b) Cl n b = ab;

( c) The ideal ab consists of nilpotent elements.

Hint: You may use the assertion of B2 (a);

( d) There exists a positive integer n such that A is isomorphic to the product ring (A/an) x (A/bn).

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Fields

Cl. (10 points) Find the minimal polynomial of ,J2 +~over the field of rational numbers Q.

C2. (15 points) Let a be a primitive 16-th root root of unity over the field F. Determine the dimension [F(a) : F] when Fis:

a) The field of 9 elements,

b) The field of 7 elements,

c) The field of 17 elements,

d) What can you say in the case p = 2?

C3. (20 points) Let F be an infinite field of characteristic p > 0. Recall that a finite dimensional extension L / F is said to be simple if L = F ( u) for some element u E L.

a) Suppose L/ F has a finite number of intermediate fields. Show that L / F is simple.

b) Let K be an intermediate field F C K C L, and suppose that L = F( u) with xr + a1xr-l + • • • + ar the monic irreducible polynomial of u over K, ai EK. Show that K = F(a 1 , a2 , ..• , ar)-

c) Conclude that L /Fis simple if and only if there are a finite number of intermediate fields.

d) Let E = F(x, y) where x and y are indeterminates, and set

Show that M / E has an infinite number of intermediate fields.

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Linear Algebra

Dl. (10 points) Let V be a finite dimensional vector space over a field F and T : V ---+ V a linear map whose characteristic polynomial has distinct roots, all in F. Show that T is diagonalizable.

D2. (15 points) Describe all non-diagonalizable 4 x 4 matrices over Q such that A5 + A2 = 0 up to similarity. Justify your answer.

D3. (20 points) Let V be a complex vector space with positive definite inner product ( , ), and T: V---+ Va linear map. Recall that the adjoint T* of T is defined by:

(T(x),y) = (x, T*(y))

for all x, y E V, and that T is called normal if T and T* commute. Suppose that T 3 = TT*. Let U be the kernel of T and W the orthogonal complement of U in V.

(a) Show that Wis T*-invariant.

(b) Show that U is T* -invariant.

( c) Conclude that W is T invariant.

( d) Show that the restrictions of T and T* to W commute.

( e) Show that T is normal.

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Spring 2001 Algebra Qualifying Exam

Everyone must do two problems in each of the four sections.

To pass at the Ph.D. level, you must attempt at least three 20-point problems. On multiple part problems, do as many parts as you can; however, not all parts count equally.

A. Groups.

Al. (10 points) Determine a complete set of groups of order eight up to isomorphism and show that every group of order eight is isomorphic to one of these.

A2. (15 points) A finite group G acts on itself by conjugation. Determine all possible G if this action yields precisely three orbits. Prove your result.

A3. (20 points) Let G be a finitely generated group.

a. Show for each integer n there exist finitely many subgroups of index n.

b. Suppose that there exists a subgroup of finite index in G. Prove that G contains a characteristic subgroup of finite index.

B. Rings.

Bl. (10 points) A commutative ring R with unit is said to be a local ring if it has a unique maximal ideal. Show that a commutative ring R with unit is a local ring if and only if for any two elements u, v E R satisfying u + v = l at least one of u, vis a unit of R.

B2. (15 points) Let R = R[x, y]. Find a finitely generated R-module M that is not a direct sum of cyclic R-modules, and prove that it is not.

B3. (20 points) Let fi(z1,••·,zn),f2(z1,••·,zn), ... ,fn(z1,••·,zn) be n polynomials in C[z1, ... , Zn]- Assume that fi(O, 0, ... , 0) = 0 for all i = 1, ... , n. Prove that the origin is the only point of en where all of the Ji vanish if and only if the ideal I generated by Ji, ... , f n contains all monomials of degree N for some sufficiently large N.

C. Fields.

Cl. (10 points) Let F be a prime field, i.e., the rationals or a field with p elements. Prove that an algebraic closure of F has infinite degree over F.

[Hint: You may want to do the two cases separately.]

C2. (15 points) Let f E Q[x] be a polynomial of degree three. Let K = Q(0) be a splitting field off. Determine all the possible galois groups of K/F, prove these are all such, and give explicit examples of K, i.e., determine a 0 or f.

C3. (20 points) Prove that the polynomial x 4 + 1 is not irreducible over any field of positive characteristic.

D. Linear Algebra.

Dl. (10 points) Let V = Mnxn(R) be the n x n real matrices. If A, B E V, define < A,B >= tr (AtB).

(a) Prove that <,>is a positive-definite symmetric inner product on V.

(b) If Ei,j is the matrix with a 1 in the i, j place and zeros elsewhere, prove that { Ei,j I 1 :s; i, j :s; n} is an orthonormal basis for V.

D2. (15 points) Let k be a field. Prove that {xi® yi I i,j 2: 0} is a basis for the k-vector space k[x] ®k k[y]. Use this to show that k[x] ®k k[y] ~ k[x, y] as vector spaces over k.

D3. (20 points) Let SO(3) = {A E Maxa(R) I At A= I and det(A) = 1}.

(a) Prove if A E SO(3) then +1 is an eigenvalue of A. (b) Prove that if W is the subspace of R 3 orthogonal to a non-zero eigenvector of

A E SO(3) with eigenvalue +1 then A takes W to W and acts as rotation by some angle () on W.


Fall 2001

Everyone must do tyVo problems in each of the four sections. To pass at the Ph.D. level, you must attempt at least three 20-point problems. On multiple part problems, do as nia.ny parts as you can; however, not all parts count equally.

Groups

Gl. {10 points) Let G be a finite group whose center has index n. Show that every conjugacy class in G has at most n element~.

G2. (15 points) Let G be a subgroup of Sn that acts transitively on the set {1, 2, ... , n}. Let H be the stabilizer in G of a.n element x E {1, 2, ... , n}. Prove that n gHg-1 = {e}

gEG

G3. {20 points) Let G be the group of ma.trices of the form \

where a E {Z/p)* a.nd a E Z/p. Describe all normal subgroups of G. Hint: find a. convenient normal subgroup of order p.

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Rings

Rl. (10 points) Let R be a commutative ring, I C R a nonzero ideal. Prove that if I is a free R-module then I= aR for an element a ER which is not a zero divisor in R. Hint: consider the rank of I.

R2. (15 points) (a) Give an example of prime ideal in a commutative ring that is not maximal.

(b) Let R be a commutative ring with identity. Suppose for every element x E R there exists an integer n = n(x) > 1 such that xn = x. Show that every prime ideal in R is maximal.

R3. (20 points) Let R be a ring.

(a) Prove that if a is a nilpotent element in a ring R with identity, then the element 1 + a is invertible.

In the next two parts, let J(X) = ao + a1X + · · · + anXn be a polynomial in R[X] of degree n, that is, an f. 0.

{b) Show that if R is an integral domain, then f(X) is invertible in R[X] if and only if n = 0.

(c) Show that if R is a commutative ring, f(X) is invertible in R[X] if and only if all ao is invertible and ai are nilpotent in R for every i 2:: 1.

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Fields

Fl. (10 points) Let f(x) = x 3 - 2x - 2.

(a) Show that f (x) is irreducible over Q.

(b) Let 0 be a complex root of f(x). Express 0- 1 as a polynomial in 0 with coefficients in Q.

F2. (15 points) Let f(x) = x3 + nx + 2 where n is an integer. Determine the (infinitely many) values of n for which f is irreducible over Q.

F3. {20 points) Let G be the Galois group of xP - 2 over Q where p is a prime. Show that G is isomorphic to the group of matrices of the form

where a E (Z/p)* and a E Z/p.

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Linear Algebra

LAL (10 points) Let T be a linear operator on a finite-dimensional vector space V such that Im(T) and Im(T 2 ) have the same dimension. Show that ker T n Im(T) = 0.

LA2. (15 points) Find all similarity classes of 4 x 4 matrices A over (Q such that A2 -:/- ±A and A2 -:/- I but A3 = A (I is the identity 4 x 4 matrix).

LA3. (20 points) Let V be a vector space over a field k. A bilinear form f : V x V ➔ k is called skew-symmetric if f (u, v) = -f(v, u) for all v, u EV and is called alternating if f ( v, v) = 0 for all v E V.

(a) Prove that every alternating form is skew-symmetric.

(b) Give an example of a skew-symmetric form which is not alternating. Hint: choose k of characteristic 2.

(c) Show that all alternating forms on V form a vector space Alt(V) and find dimAlt(V) if dim V = n.

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Algebra Qualifying Exam (Spring 2002)

Test Instructions: All problems are worth 20 points. Your total score will be computed by dropping the the lowest scoring problem. In problems where arguments must be given, you will lose points if you fail to state clearly the basic results you use.

GROUP THEORY

PROBLEM 1.

Show that a group G of order 2m, where m odd, has a normal subgroup of order m.

PROBLEM 2.

List, up to isomorphism, all finite abelian groups A satisfying the following two conditions:

( i) A is a quotient of Z2, and

(ii) A is annihilated by 18, i.e. 18a = e for all a in A.

Your list should contain a representative of each isomorphism class exactly once. How many groups are there?

PROBLEM 3.

Prove that a group G of order 120 is not simple.

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RING THEORY

PROBLEM 1.

Let R be a ring and A and B be two non-isomorphic simple, left R-modules (a left-module is simple if it has no proper submodules, i.e., submodules other than {0} and itself). Show that the only proper submodules of M = AEBB are {(a,0): a EA} and {(0,/3): /3 EB}.

PROBLEM 2.

Let R be a commutative local ring, that is, R has a unique maximal ideal M.

(i) Show that if x lies in M, then 1 - x is invertible.

(ii) Show that if R is Noetherian and I is an ideal satisfying 12 = I, then I= 0. Hint: consider a minimal set of generators for I.

PROBLEM 3.

Let F2 be the field with 2 elements and let R = F2[X]. List, up to isomorphism, all R-modules of order 8.

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LINEAR ALGEBRA

PROBLEM 1.

Let 'P : M3(Q) -----t M3(Q) be the map sending m to 'P(m) = m2 + 3m + 3. Show that 'P(m) i= 0 for all m E M3(Q).

PROBLEM 2.

Let A be a real matrix with column vectors A1, A2, ... , An. If the Aj are mutually orthogonal, then

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I detAI = II IAjl j=l

This follows because I det (t A · A) I = I det A I 2 and t A · A is a diagonal matrix with diagonal entries IA112, IA212, ... , IAnl2 . Prove that a general matrix satisfies the inequality

n

I detAI :S II IAjl j=l

Hint: apply the Gram-Schmidt orthogonalization process to the columns.

PROBLEM 3.

Let T E M3(C) and let Ar be the centralizer of T in M3(C). Show that dim(Ar) ~ 3 and describe (up to similarity) the linear transformations T such that dim(Ar) = 3.

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GALOIS THEORY

PROBLEM 1.

Let F7 be the field with 7 elements and let L be the splitting field of the polynomial X 171 - 1 = 0 over F7. Determine the degree of L over F7,

explaining carefully the principles underlying your computation.

PROBLEM 2.

Show that there exists a Galois extension of Q of degree p for each prime p. State precisely all results which are needed to jusfify your answer.

PROBLEM 3.

Let a= Ji+ 2 where i = -J=I. (a) Compute the minimal polynomial of a over Q.

(b) Let F be the splitting field and compute the degree of F over Q;

(c) Show that F contains 3 quadratic extensions of Q;

(d) Use this information to determine the Galois group.

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Winter 2002

Everyone must do two problems in each of the four sections. If three problems of a section are tried, only two problems of highest score count (the lowest score is ignored). On multiple part problems, do as many parts as you can; however, not all parts count equally.

Groups Al. Let G be a free abelian group of rank n for a positive integer n

(therefore G ~ zn as groups). (a) Prove for a given integer m > 1, there are only finitely many

subgroups H of index m in G; (b) Find a formula of the number of subgroups of G of index 3.

Justify your answer. A2. Prove or disprove: there exsits a finite abelian group G whose

automorphism group has order 3. A3. Let Sand G be p-groups (with G -=f { e} ), and assume that S acts

on G by automorphisms. Show that the fixed subgroup G8 = {g E Gls(g) = g for all s E S} is non-trivial (i.e., is not the trivial subgroup { e}).

Rings Bl. Let F be a field and A be a commutative F-algebra. Suppose A

is of finite dimension as a vector space of F. (a) Prove all prime ideals of A are maximal. Hint: consider maps

R/ P-* R/ P (P prime) of the form x-* ax with a in R. (b) Prove that there are only finitely many maximal ideals of A.

B2. Let A = Mn(F) be the ring of n x n matrices with entries in an infinite field F for n > 1. Prove the following facts: (a) There are only 2 two-sided ideals of A; (b) There are infinitely many maximal left ideals of A. Hint: show

that Ax= Ay (x, y EA) if and only if Ker(x) = Ker(y). B3. Let IF 2 be the field with 2 elements and A = IF 2 [T, ~] for an inde

terminate T. Prove the following facts: (a) The group of units in A is generated by T. (b) There are infinitely many distinct ring endomorphisms of A. (c) The ring automorphism group Aut(A) is of order 2.

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Fields Cl. The discriminant of the special cubic polynomial f(x) = x 3 +ax+b

is given by -4a 3-27b 2• Determine the Galois group of the splitting field of x 3

- x + l over (a) IF3 , the field with 3 elements. (b) IF 5 , the field with 5 elements. ( c) Q, the rational numbers.

C2. A field extension K /Q is called biquadratic if it has degree 4 and if K = Q( y'a, Jb) for some a, b E Q. (a) Show that a biquadratic extension is normal with Galois group

Gal(K/Q) ~ Z/2Z x Z/2Z and list all sub-extensions. (b) Prove that if K /Q is a normal extension of degree 4 with

Gal(K /Q) ~ Z/2Z x Z/2Z then K /Q is biquadratic. C3. Let K be a finite extension of the field F with no proper interme

diate subfields. (a) If K/ Fis normal, show that the degree [K; F] is a prime. (b) Give an example to show that [K; F] need not be prime if

K / F is not normal, and justify your answer.

Linear Algebra

Dl. Let J = ( !}_1 ~) where I is then x n identity matrix. Suppose

that Sis a 2n x 2n symplectic matrix, meaning that Sis real and satisfies tSJS = J, where t5 is the transpose of S. (a) Show that t5 is symplectic. (b) Show that S is similar to s- 1 .

( c) It is always true that det S = 1. Prove this in case n = 1. D2. Suppose that A is a linear operator on the vector space ccn and

that v E ccn satisfies (A - aI)2v = 0 for some a E CC, so that v is a generalized eigenvector of A with eigenvalue a. Suppose that lal < 1. Show that

IIAmvll --+ 0

as m--+ oo, where 11-11 is the Euclidean norm on ccn. D3. Let then x n matrix A be defined over the field F. Suppose that

A has finite order: Am=I

for some positive integer m. (a) If the characteristic of Fis 0, show that A may be diagonalized

over F. (b) Show that the conclusion of (a) is not true for an arbitrary

field F.

Algebra Qualifying Exam - Fall 2002

Test Instructions: All problems are worth 20 points. You are expected to do two problems in each of the four sections. Your total score will be computed by dropping the lowest scoring problem in each section. In problems where arguments must be given, you will lose points if you fail to state clearly the basic results you use.

GROUP THEORY

PROBLEM 1.

a) Let A be a free abelian group of rank n. If H is a subgroup of A, show that His free abelian ~ rank n if and only if A/ H is finite.

PROBLEM 2.

Let G be a finite group of order 108. Show that G has a normal subgroup of order 9 or 27.

PROBLEM 3.

Let G be a finite group and P a p-Sylow subgroup. Let Na(P) be the normalizer of P in G. Show that:

- a) P is the unique p-Sylow subgroup of Na(P) (Do not quote a theorem that this is true!)

b) Na(P) is self normalizing in G

RING THEORY

-PROBLEM 1.

Let R be a commutative ring with 1, and let S = R[x] be the polynomial ring in one variable. Suppose Mis a maximal ideal of S. Prove that M cannot consist entirely of 0-divisors.

Hint: You may want to distiguish the cases x E M or x ¢:. M

PROBLEM 2.

Let R be a commutative ring with 1, and suppose I and J are ideals of R so that: J + J = R. Show that:

(i) I J =In J

(i) R/ I J ~ R/ I ffi R/ J

PROBLEM 3.

Let R be a commutative ring with 1, and let S = R[x] be the polynomial ring in one variable. Let f E S. If f is a unit of S (that is, f is invertible in S), show that f has the form f = u + g

-here u is a unit in R and g E S is a nilpotent element without constant term.

FIELDS

-PROBLEM 1.

a) Determine the minimal polynomial of u = J3 + 2-/2 over Q. b) Determine the minimal polynomial of u- 1 over Q.

PROBLEM 2.

a) Let F be the field generated by the roots of the polynomial X 6 + 3 over Q. Determine the Galois group of F / Q.

b) Describe all subfields of F.

PROBLEM 3.

Let p be a prime integer such that p 2 or 3 (mod 5). Prove that the polynomial

1 + x + x 2 + X 3 + x 4

• irreducible over Z/pZ.

LINEAR ALGEBRA

-PROBLEM 1.

Let T be a linear operator on a finite-dimensional vector space V such that im T = im T 2. Prove that ker T = ker T 2. (im and ker refer to the image and kernel of T)

PROBLEM 2.

Determine, up to similarity, all 3 x 3 matrices A over Q such that A2 + 2A3 + A 4

A+A 2 -/= 0.

PROBLEM 3.

Let T1, T2, ... , Tm be linear operators on a vector space of dimension n. Assume that

(i) dim im(Ti) = 1 and

(ii) T/ -/= 0 and TiTj = 0 for every i -/= j . • rove that m :S n.

0 but


Winter 2003

Everyone must do two problems in each of the four sections. If three problems of a section are tried, only two problems of highest score count (the lowest score is ignored). On multiple part problems, do as many parts as you can; however, not all parts count equally.

Groups

Al. List, up to isomorphism, all abelian groups A which satisfy the following three conditions: (i) A has 108 elements;

(ii) A has an element of order 9; (iii) A has no element of order 24.

A2. Let N 2: 1 be a positive integer. Show that a finitely generated group G has only finitely many subgroups of index at most N.

A3. Let N 2: 2 be an integer. Show that a subgroup of index 2 in SN is AN. Here SN and AN are the symmetric and alternating groups for N, respectively.

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2

Rings

Bl. Give an example of two integral domains A and B which contain a field F such that A® F B is not an integral domain. Justify your answer. Hint: Take A to be the field of rational functions IFp(X) for the field IF P with p elements.

B2. Let IFq be the finite field of q elements, and put F = IFq and K = IF q2. Write a : K -+ K for the field automorphism given by xu = xq. Let

B = { ( db" at) la, b EK} for a given d E px. Prove the following three facts: (a) Bis a subalgebra of dimension 4 over F inside the F-algebra

of 2 x 2 matrices over K. (b) B is a division algebra if and only if there exists no c E K

such that d = ccu. ( c) B cannot be a division algebra.

B3. Let A be a discrete valuation ring with maximal ideal M, and define

B = { ( a, b) E A x A I a - b mod M} . Prove the following facts: (a) B has only one maximal ideal; (b) B has exactly two non-maximal prime ideals.

3

Fields

Cl. Let IF q be the finite field of q elements. Answer the following questions: (a) List all subfields of IFP6 for a prime p. Justify your answer.

(b) Find a formula for the number of monic irreducible polyno-mials of degree 6 in IFp[X]. Justify your answer.

C2. Let K / F be a quadratic extension of fields and M / F be a Galois extension over F containing K such that Gal(M/ K) is a cyclic group of odd prime order p. Answer the following two questions: (a) Determine the possible groups Gal(M / F) up to isomorphisms,

and justify your answer. (b) Find the number of intermediate fields L between F and M

with [L: F] = p. Justify your answer.

C3. Find the degree of the splitting field E of X 6 - 3 over the following

fields: (a) Q[y1=3] (Q: the field of rational numbers); (b) IF 7 , the field with 7 elements; (c) IF5 , the field with 5 elements, and justify your answer.

4

Linear Algebra

Dl. Let L be the subgroup of 'll} with generators (3, 2, 1) and (2, 2, 6). Represent the quotient group A = Z3

/ L as a product of cyclic groups.

D2. List suitably the Jordan canonical forms of all matrices A that satisfy (i) tr(A) = l;

(ii) tr(A) 2 = 2det(A) + tr(A 2).

D3. Let M be a matrix with complex entries. Deduce, using the structure theory of modules, that: (i) M = S + N, where Sis semisimple (i.e. diagonalizable), N is

nilpotent, and N and S commute. (ii) N = P(M) where P(X) is a polynomial with complex coeffi

cients.

Algebra Qualifying Exam - Fall 2003

Test Instructions: All problems are worth 20 points. You are expected to do two problems in each of the four sections. Your total score will be computed by dropping the lowest scoring problem in each section. In problems where arguments must be given, you will lose points if you fail to state clearly the basic results you use.

GROUP THEORY

PROBLEM 1.

Let G be a finitely generated group, and n > 1 an integer. Show that G has at most a finite tiumber of subgroups of index n.

PROBLEM 2.

Let G be a finite group, K a normal subgroup, and P a p-Sylow subgroup of K for some prime p. Prove that G = KNa(P).

PROBLEM 3.

Suppose G is the free abelian group on generators x, y, z, w, considered as an additive group. Let a = x - z + 2w, b = x - y + w, c = 3x - y - 2z + 5w, d = 2x - 2y + 4w. If H = (a, b, c, d),

9-etermine the structure of G / H.

RING THEORY

PROBLEM 1.

a) Let R be a commutative ring with 1. Suppose f E R[x] is a non-zero 0-divisor in the polynomial ring R[x]. Assume that R has no non-zero nilpotent elements. Show there is a non-zero element a ER so that a· f = 0.

b) Give an example of an R and f so that all coefficients of f are 0-divisors in R, but f is not a 0-divisor in R[x].

PROBLEM 2.

Let R be a ring, not necessarily commutative, and M a Noetherian left R-module. Suppose f: M ➔ Mis a surjective R-module map from M to M. Prove that f is an isomorphism.

PROBLEM 3.

a) Let R be a commutative ring with 1, and S a multiplicatively closed subset of R not containing 0. Suppose I is an ideal of R maximal with respect to exclusion of S (i.e. In S is empty and J is largest such). Prove that J is a prime ideal of R.

b) Show that every prime ideal of R arises as in part a). -

FIELDS

PROBLEM 1.

Determine the Galois group of the polynomial X 4 + 3X 2 + 1 over Q.

PROBLEM 2.

-et f(X) be a polynomial of degree n > 0 over a field F.

a) Prove that there is a field homomorphism a: F(X)-+ F(X) such that a(X) = f(X).

b) Let L be the image of a. Prove that the field extension F(X)/ Lis finite and find its degree.

c) Find the minimal polynomial of X over L.

PROBLEM 3.

Let p be a prime integer. Suppose that the degree of every finite extension of a field F is divisible by p. Prove that the degree of every finite extension of F is a power of p.

LINEAR ALGEBRA

PROBLEM 1.

Let A be a linear operator in a finite dimensional vector space. Prove that if A 2

Trace(A) = Rank(A).

PROBLEM 2.

A then

Let L / F be a field extension and let A and B be n x n matrices over F. Prove that if A and B are conjugate in Mn(L), then A and Bare conjugate in Mn(F).

PROBLEM 3.

Let V be a finite dimensional vector space over a field F. Let B be a non-degenerate bilinear form on V. (For every nonzero v EV there is u EV such that B(u, v) =f. 0.)

a) For every v E V define a linear form lv : V ----+ F by lv(u) = B(u, v). Prove that the map V ----+ V* given by v H lv is an isomorphism of vector spaces.

b) Prove that for every linear operator a of V there is a linear operator a* such that B(a(u), v) = B(u, a*(v)) for all u, v EV.

c) Prove that (ab)* = b*a* for every two linear operators a and b.

d)

-Suppose that B is either symmetric or skew-symmetric ( that is B ( u, v) B(u,v) = -B(v,u) respectively). Prove that a**= a for a linear operator a.

B(v, u) or

Algebra Qualifying Exam (Spring 2004)

Test Instructions: Everyone must do two problems in each of the four sections. If three problems of a section are tried, only the two problems of highest score count (the lowest score is ignored). On multiple part problems, do as many parts as you can; however, not all parts count equally.

GROUP THEORY

PROBLEM 1.

A group G is said to act transitively on a set S if for any element s E S, then

S=Gs.

Suppose G is finite and that G acts transitively on S. Let f(g) be the number of elements of S fixed by the action of g E G on S. Prove

IGI = Lf(g). gEG

PROBLEM 2.

Classify all groups of order 2 · 7 • 11.

PROBLEM 3.

Let G be a finite group and H a subgroup of G. Let n = ( G : H) be the index of Hin G.

(a) Show that

is a factor of n!.

(b) Suppose that the index ( G : H) is the minimal prime factor of the order of G. Show His a normal subgroup.

1

RING THEORY

PROBLEM 1.

Let R be a commutative noetherian ring with unity 1 and f : R ----+ R a surjective ring homomorphism, i.e. f(R) = R. Show f is an isomorphism.

PROBLEM 2.

Let R be the ring Q[x] and let M be the submodule of R 2 generated by the elements (1 - 2x, -x 2 ) and (1 - x, x - x2 ). According to the theory of modules over principal ideal domains, R 2 / M is a direct sum of cyclic R modules of the form R/ P( x) for manic polynomials P( x). Find such a direct sum decomposition explicitly in this case.

PROBLEM 3.

Suppose we are given a collection of polynomials in r variables with rational coefficients:

We define the complex algebraic set Ve c er by

Suppose Ve is not empty. Show that there is a finite extension K of Q and a point

with all ak E K.

2

LINEAR ALGEBRA

PROBLEM 1.

(a) For which z EC is

( 1 2z) z-1 1

not similar over C to a diagonal matrix? Justify your answer.

(b) Let Jn be the n x n matrix each of whose entries is 1. Determine those n E z+ for which Jn is diagonalizable over C and give a diagonal matrix that is similar to Jn for such n.

PROBLEM 2.

Find an explicit formula for the determinant of a 3 x 3 complex matrix A as a polynomial in the traces tn = Tr(An) for n = 1, 2, ....

PROBLEM 3.

Let V be a vector space over C of dimension d > 0. Suppose that A, B, C are linear operators on V such that

AB-BA= C.

Suppose also that C commutes with both A and B. If V has no proper non-zero subspace that is left stable under all three operators, show that d = l.

3

FIELD THEORY

PROBLEM 1.

Let K be a finite extension of (Q obtained by adjoining to (Q a root of f(x)=x 6 +3.

(a) Show that K contains a primitive 6-th root of unity.

(b) Show that K is a Galois extension of (Q.

(c) Determine the number of fields F of degree 3 over (Q with F CK.

PROBLEM 2.

Suppose that f(x) is a polynomial in (Q[x] of degree d > 1 with d roots x1, ... , xd in C. If x2 = ax1 for a E (Q different from -1, prove that f ( x) is reducible.

PROBLEM 3.

Let· K be a field and L a finite extension of K. Consider the set A of all elements x E L with the property that K[x] is a Galois extension of K with an abelian Galois group Gal(K[x]/K). Show that A is a subfield of L containing K .

4

1. Groups

Algebra Qualifying Exam Fall 2004

(a) Is there a finite group G such that G / Z ( G) has 143 elements? (Z(G) is the center of G.)

(b) Prove that every group of order 30 has a subgroup of order 15.

(c) Find all finite groups that have exactly three conjugacy classes.

2. Rings

(a) Let X be a finite set and let A be the ring of all functions from X to the field R of real numbers. Prove that an ideal M of A is maximal if and only if there is an element x E X such that

M = {J E Alf(x) = O}.

(b) Describe all n E Z such that the ring Z/nZ has no idempotents other than 0 and 1.

(c) A (non-commutative) ring R is called local if for every a E R either a or 1 - a is invertible. Prove that non-invertible elements of a local ring form a (two-sided) ideal.

3. Linear Algebra

(a) Determine whether it is true in general that in GLn(C) every matrix is conjugate to its transpose.

(b) Determine the number of conjugacy classes in GL3(C) whose elements A satisfy the polynomial X 2 - 2X + 1 = 0.

(c) Let A be an n-by-n symmetric matrix with real entries and let, for j E [1,, n], Aj be the submatrix consisting of the entries of A in the first j rows and columns of A. Show A is positive definite iff det(Aj) > 0 for all j E [1,,n].

4. Fields

(a) Let a be an integer and let p be a prime. Show that if a is not a p-th power, then XP - a is irreducible over Q.

1

(b) Show that if K and L are finite separable extensions of F with K Galois over F, such that Kn L = F, then [KL:F]=[L:F][K:F]. Show that if neither K nor L are Galois over F, then this fact need not be true.

(c) By using several quadratic extensions of the rational function field in two variables F = F 2 (X, Y) where F2 is the field with 2 elements, give an example of a field extension of finite degree of F that possesses infinitely many intermediate fields.

2


Winter 2005

Test Instructions: Everyone must do two problems in each of the four sections. If three problems of a section are tried, only the two problems of highest score count ( the lowest is ignored). For multiple part problems, do as many parts as you can; however, not all parts count equally.

Groups

Al. (a) If G is a simple group that has subgroup of index n, prove that the order of G is a factor of n!.

(b) Prove that there is no simple nonabelian group of order pem with e > 0 for a prime p > m.

A2. An additive abelian group is called divisible if multiplication by n for every positive integer n is a surjective endomorphism. ( a) Show that if G is divisible, G / H is divisible for any subgroup

Hof G. (b) Give an example (with a proof) of a divisible group for which

the multiplication by n is not an automorphism for every positive integer n.

( c) Prove or disprove that there is only one isomorphism class of finitely generated divisible groups.

A3. Let G be a free abelian group of finite rank r. (a) Show that there are only finitely many homomorphisms of G

into 'lL/n'lL for each positive integer n. (b) Find a formula of the number of surjective homomorphisms

of G onto 'lL/p'lL for a prime p if r = 2.

Linear Algebra

Bl. Ann x n real symmetric matrix Pis positive definite if the inner product P(x, y) = txPy is positive definite (that is, P(x, x) > 0 for all 0 -::p x E ~n). Let S be an n x n invertible real symmetric matrix. Let W C ~n be a subspace such that the inner product S(x, y) = txSy is positive definite on W but S is not positive definite on W + ~x for any x (/_ W. (a) Show that ~n =WEB W_1_ for W_1_ = {x E ~nlS(x, W) = 0}. (b) For each x E ~n, writing x = xw EB xw1- for xw E W and

xw1- E W_1_, defineP(x,y) = S(xw,Yw)-S(xw1-,Yw1-). Show that ps- 1 = sp-l and pis positive definite.

1

2

(c) If Pis symmetric positive definite and satisfies ps- 1 = sp-1, there exists a subspace W such that P =Son Wand P = -S on w1-.

B2. Let V be a two dimensional vector space over a field F. Let T : V --+ V be a linear transformation of finite order m. Prove the following facts: (a) If F = Q, then m :S 6. (b) For any given positive integer N, there exists a finite field F

and a nondiagonalizable T of order greater than N. B3. Let V be a finite dimensional vector space over a field F and

T : V --+ V be a linear transformation. Let v E V be a non-zero vector in V. Prove the following facts: (a) There exists a monic polynomial P(X) in F[X] such that

P(T)v = 0. (b) Among monic polynomials P(X) E F[X] with P(T)v = 0,

there exists a unique polynomial P0 (X) of minimal degree. (c) If P(T)v = 0, then P0(X) is a factor of P(X) in F[X].

Rings Cl. Let R be an integral domain. If m is a maximal ideal in R, view

the localization Rm := s-1 R, with S = R\ m, in the quotient field of R. Show that

R= n mEMax(R)

C2. Let R be a commutative Artinian ring. Show that there are only finitely many prime ideals in R and every one of them is maximal.

C3. Let R s;;: A s;;: B be commutative rings. Suppose that R is noetherian and B is a finitely generated R-algebra. Suppose that as an A-module B is finitely generated. Show that A is a finitely generated R-algebra.

Fields Dl. Show that the identity map is the only field automorphism of the

real numbers. Show this is not true of the complex numbers. D2. Let F be a field of positive characteristic p and J the polynomial

xP - x - a E F[x]. Let K/F be a splitting field of J. Show that K / F is galois and determine explicitly ( with proof) the Galois group of K/ F.

D3. Let K / F be a finite extension of finite fields. Prove that the norm map N K/F : K --+ F is surjective.


Test Instructions: All problems are worth 20 points. You are expected to do two problems in each of the four sections. Your total score will be computed by taking the two best scoring problems in each section. In problems where arguments must be given, you will lose points if you fail to state clearly the basic results you use.

(1) Groups

(a) Let G be an abelian group generated by n elements. Prove that every subgroup of G can also be generated by n elements.

(b) Let N be a normal subgroup of G. Prove that for a Sylow psubgroup P of G, the intersection P n N is a Sylow p-subgroup of N.

( c) Is there a nontrivial action of the alternating group A4 on a set of two elements?

(2) Rings

(a) Let J and J be ideals of a commutative ring R with unit such that J + J = R. Prove that J • J =In J.

(b) Prove that the factor ring R[x, y]/(y 2 -x 3)R[x, y] is not a P.I.D. (c) Let x,y,z,t be elements of a (non-commutative) ring R such

that xz = yt = l, xt = yz = 0 and zx + ty = l. Prove that the left R-modules Rand R EB Rare isomorphic.

(3) Linear Algebra

(a) Prove that if three distinct real numbers Ai and three arbitrary numbers µj are given, then there exists a unique polynomial f(x) E R[x] of degree at most 2 such that f(Ai) = µi,

(b) Let K / F be a field extension of finite degree n and assume K = F(a) where a satisfies a polynomial f(x) of degree n in F[x]. Let cp : K -+ K be the F-linear map cp(x) = ax. Show that the eigenvalues of cp coincide with the roots of f(x).

(c) A bilinear form A(x, y) on a vector space V over C is called alternating if A(v, w) = -A(w, v) for all v, w E V and it is called non-degenerate if, for each nonzero v E V there exists w EV such that A(v, w) =/:-0.

(i) Prove that any two non-degenerate alternating forms on V = C2 differ by a scalar multiple.

(ii) Does this remain true for V = en for n > 2? Either prove true or give a counterexample.

1

2

( 4) Fields

(a) Let F q be the finite field with q = pn elements and let N : F q ----+ F p be the norm map, defined by

Nx = IT a(x)

where a runs over the Galois group G =Gal(F q/F p)- Prove that N is surjective.

(b) Let cp(x) = x4+a3x 3+a2x 2+a1x+ao be an irreducible polynomial of degree 4 in Q[x] and let K be the field generated by the complex roots a1, a2, a3, a4 of cp. Let F be the field generated by:

/31 = (a1 + a2)(a3 + a4), /32 = (a1 + a3)(a2 + a4),

/33 = (a1 + a4)(a2 + a3).

Prove that K / F is an abelian extension, that is, the Galois group H =Gal(K/ F) is abelian. Hint: prove that the /3i are distinct and determine the possible elements of Gal(K/Q) that fix them.

(c) Let K be the splitting field of cp(x) = x 11 - 7 over Q. Describe the Galois group G =Gal(K/Q) by giving generators and relations. Determine the number of quadratic subfields of K ( a quadratic subfield is a subfield ECK such that [E: Q] = 2).

Algebra Qualifying Exam Spring 2006

Test instructions: All problems are worth 20 points. You are expected to do two problems from each of the four sections. Your total score will be computed by taking the two best scoring problems from each section. In problems where an argument must be given, you will lose points if you fail to state clearly the basic results you use.

Groups

G 1. Let G be a finite group. Let K be a normal subgroup of G and P a p-Sylow subgroup of K. Show that

G = KNc(P).

G2.

(a) What is the order of SL 2 (F4)?

(b) Show there is an isomorphism from SL 2 (F4 ) to A5 .

Hint: Consider the action of SL 2 (F4 ) on the set of one dimensional subspaces of the vector space ( F4 ) 2 of dimension two over F4 .

G3. Let G be a group of order 2000 and suppose that P and P' are two distinct Sylow 5 subgroups of G. Let

I= PnP'.

(a) Prove that III = 25.

(b) Show that the index of Nc(I) is at most 2.

Rings

Rl. Suppose D is an integral domain and suppose that D[x] is a principal ideal domain. Show D is a field.

R2. Let R be a commutative Noetherian ring with unit, and suppose M is a finitely generated R module. Suppose f : M -r M is an R module homomorphism which is onto. Show that f is an isomorphism.

R3. Let R be a commutative ring with unit and m a maximal ideal of R.

(a) Suppose I 1 ... In are ideals of R and that

where Ii ... In is the product of the ideals. Show

for some k.

(b) Suppose that R satisfies the descending chain condition (dee) on ideals, i.e. every strictly decreasing sequence of ideals is finite. Show R has only a finite number of maximal ideals. You may use part (a), but not theorems on the structure of rings satisfying the dee.

Fields

Fl.

(a) Show that the Galois group of the splitting field of

X 4 -2

over Q has order 8.

(b) Is this Galois group isomorphic to the dihedral group, the quaternion group or one of the three abelian groups of order 8?

F2. Let F be a finite field.

(a) Show that more than half the elements of F are squares.

(b) Show that every element of F is the sum of two squares.

F3. Let K be a finite extension of the field F with no proper intermediate fields.

(a) If K/ Fis normal, show [K: F] must be prime.

(b) Give an example to show that [K: F] need not be prime if K/F is not normal, explaining why your example works.

Linear Algebra

Ll. Let A be a 2 x 2 complex matrix and let WA be the space of all 2 x 2 matrices that commute with A.

(a) What is the minimal possible dimension of WA as A varies over all 2 x 2 complex matrices?

(b) Classify those A such that WA has minimal dimension.

L2. Let A be a 3 x 3 matrix over a field F that satisfies

A 4 = I and A2 -f I,

where I is the identity matrix. Find all similarity classes of such A when

(a) F = Q

(b) F is the field of two elements.

L3. Let T1 , T2 , ... Tn be linear operators on a vector space of dimension m

over a field F. Assume that

(a) dimim(Ti) = 1 for each i and

(b) T/ -f O and TiTi = 0 for i -f j. Show

'

I


Fall 2006

Test Instructions: Everyone must do two problems in each of the four sections. If three problems of a section are tried, only the two problems of highest score count (the lowest is ignored). For multiple part problems, do as many parts as you can; however, not all parts count equally.

Groups

Gl. List all finite groups G whose automorphism group has prime order. Justify your answer.

G2. Let G be a finite group and H be a non-normal subgroup of G of index n > 1. (a) Show that if \H\ is divisible by a prime p ~ n, then H

cannot be a simple group; (b) Show that there is no simple group of order 504 = 23 • 32 • 7.

(Hint: Choose a good prime /!,, and let G act on the set of Sylow /!,-subgroups getting an embedding of G into a permutation group, and apply (a). You may use the fact that the alternating group An is simple if n ~ 5.)

G3. Let

SL2('ll/p'll) = { ( ~ ~) la, b, c, d E 'll/p'll, ad - be= 1} , where p is an odd prime. (a) Prove that any subgroup of a cyclic group is cyclic. (b) Compute the order of G. ( c) Prove that for any odd prime /!,, the Sylow /!,-subgroup of G

is cyclic. (Hint: You may use the fact that the multiplicative group px of a finite field F is cyclic).

Rings

Rl. Determine all prime ideals in the polynomial ring 'll[x]. Justify your work.

R2. Let R be a noetherian domain. A non-zero element x in R is called a prime element if (x) is a prime ideal. Prove all of the following: (a) Every nonzero non-unit in R is a product of irreducible

elements. (b) Every nonzero ideal J =f R in R contains a (finite) product

of non-zero prime ideals. 1

2

I

( c) If every nonzero prime ideal in R contains a prime element then every irreducible element in R is a prime element.

[You may not use theorems about UFD's] R3. Let R be a commutative ring and M a finitely generated R

module. Suppose there exists a positive integer n and a surjective R-module homomorphism c.p : M - R'. Show that ker c.p is also a finitely generated R-module.

Fields Fl. Let F be a finite field of positive characteristic p. Show that

the unit group F \ {O} of F is a cyclic group and that F is a Galois extension of 7l/p7l.

F2. Let f(x) be the polynomial x6 + 3 over (Q (the field of rational numbers). Determine the Galois group of f(x), i.e., the Galois group of K/(Q where K is a splitting field of f(x).

F3. Let f(x) be an irreducible polynomial over F and K/ Fa normal extension. Show that f(x) factors into irreducible polynomials over K all of the same degree.

Linear Algebra 11. Let V be an n-dimensional vector space over a field F (for

finite n) and CY : F - F be a field homomorphism (sending 1 to 1). We regard V 0F,a F as an F-vector space via the Fmultiplication given by a(v 0 a:) = (av) 0 a: for a, a: E F and VE V. (a) Compute the formula of dimp(V 0F,a F) if F is a finite

extension of a field k fixed by CY, where V 0 F,a F is the tensor product over F regarding Fas F-module via CY.

(b) Let p be a prime. If F = lFp(x1, ... , Xm) (lFp = 7l/p7l) and CY ( </>) = ¢P for all </> E F, compute the formula of dimp(V0F,aF). Here lFp(x1, ... , Xm) is the field of fractions of the polynomial ring lFp[x1, ... , xm] of m variables.

(c) Give an example of a field F of characteristic 3 and a homomorphism CY : F - F such that dimp V 0 F,a F = 2 dimp V, and justify your example.

L2. Let V be a finite dimensional vector space over the rational number field (Q and I : V x V - Q be an alternating bilinear map (that is, I(x, y) = -I(y, x) for all x, y E V). We call I degenerate if there exists nonzero x E V such that I(x, V) = 0, and J is non-degenerate if I is not degenerate. (a) Show that if Vis two dimensional and I is non-degenerate

then there exists a basis x, y such that J ( x, y) = 1.

. .

I

3

(b) Show that J is degenerate if dimQI V is odd. ( c) If diIDQ! V = 2m is even and J and J' are two nondegenerate

alternating forms on V, show that there exists an invertible linear transformation T : V ---+ V such that J' ( x, y) = I(Tx, Ty) for all x, y EV.

13. Let T: V---+ V be a linear transformation on an n-dimensional vector space V over (C with n 2 1. Suppose that yn = 0 but yn-l-=/= Q.

(a) Compute dim Ker(T 101).

(b) If S : V ---+ V is a linear transformation with ST = TS and dimS(V) = n - 2, compute dimS 101 (V).



Groups

1. Let G be a simple group containing an element of order 21. Prove that every proper subgroup of G has index at least 10.

2. Find the number of subgroups of '11.P of index 5.

3. Let G be a group with cyclic automorphism group Aut(G). Prove that G is abelian.

Rings

1. Let D be a division ring ( a ring with identity in which every nonzero element is invertible). Let R = Matn(D) be the ring of n x n matrices with entries from D. Prove that R has no two-sided ideals other than R itself and { 0}.

2. Let R = End(V) be the ring of all linear endomorphisms of an infinite dimension complex vector space V with countable basis { e1, e2, ... } . Prove that Rand R E£l Rare isomorphic as left R-modules.

3. (a) Give a description of all maximal ideals of the ring C[x, y]. Justify your description. You may use the Nullstellensatz.

(b) Let M = (x2 -y, y 2 - 5) be an ideal in R = Q[x, y]. Prove that M is a maximal ideal.

2

Fields

1. Let F = Q ( () where ( = e21ri/ 5 and let E / F be a Cyclic Galois extension of degree 5. Prove that there exists a E F such that E = F( lfci). Hint: find a E E such that a(a) = (a, where a is a generator of the Galois group Gal(E / F).

2. Let K = Q( v'3, V5). (a) Prove that K has only one subfield F c K such that

[F:Q] =2. (b) Find all subfields of K. (c) Find an element u EK such that K = Q(u). (d) Describe all elements u EK such that K = Q(u).

3. Let F = Z/3. First explain why F[x]/(x 2 - 2) is isomorphic to F[x]/(x 2 - 2x - 1). Then find an explicit isomorphism:

</>: F[x]/(x 2 - 2) -+ F[x]/(x 2 - 2x - 1).

Linear Algebra

1. Let A be a linear operator in a Q-vector space V of dimension n such that the minimal polynomial of A has degree n. Prove that every linear operator on V that commutes with A is a polynomial in A over Q.

2. Let G = GLn(C) be the multiplicative group of invertible n x n matrices over C. Prove that every element of finite order in G is conjugate to a diagonal matrix.

3. Let A(x, y) be a bilinear form on a vector space V of finite dimension and

Vi={xEV suchthat A(x,y)=O forall yEV},

\1r = {y E V such that A(x, y) = 0 for all x EV}. Prove that dim Vi= dim Vr.

•


Test Instructions: Each problem is worth 20 points. You must attempt to do at least two problems in each of the four sections. Your total score will be computed using only the two best scoring problems in each section. Whether you pass or fail depends on your performance in each section, not only on the total score. In each problem you may lose points if you do not explain clearly your reasoning or any theorems which you quote.

Groups G 1. Let F be a finite field of characteristic p and let G be a subgroup

of order pa of the group GL(N, F) of invertible Nby N matrices with entries in F. Show that there is a non-zero vector v in pN such that gv = v for every g E G.

G2. Let M be the submodule of Z3 generated by elements (0, 3, 2), (6, 48, 24) and (6, 24, 12). Describe the quotient group Z3 /M by giving a product of cyclic groups to which Z3 / M is isomorphic.

G3. A group G acts doubly transitively on a set X if for each pair (xi, x2) and (Yi, Y2), with xi #-X2 and Yi #-Y2 of pairs of points of X, there exists a g E G such that gxi = Yi and gx2 = Y2• Show that if a finite G acts non-trivially and doubly transitively on X, then the stabilizer Bx of any point x in X is a maximal proper subgroup of G. (Here a subgroup M of G is maximal proper if M is not equal to G and if, for any subgroup H of G which contains M, either H = M or H = G holds. )

Rl.

R2.

R3.

Rings Let F be a field and A be a commutative F-algebra. Suppose A is of finite dimension as a vector space of F. (a) Prove that if A is a domain, A is a field. (b) Prove that even if A is not a domain, there are only finitely

many prime ideals of A. Let A be a commutative ring with identity, and write V for the set of all prime ideals of A. Put D(x) = {P E Vix (/. P} for x EA. Prove (a) D(a) = D(an) for integers n > 0; (b) V = D(a) U D(b) U D(c) if a3 + b3 + c3 is invertible in A.

Determine all isomorphism classes of modules over the polynomial ring lF2 [X] which are of dimension 2 over lF2 , and justify your answer. Here lF 2 is a field of two elements.

1

2

Fields Fl. Let F be a field. Show that the unit group F \ {0} of F is

finitely generated if and only if F is finite. F2. Let f(x) be the polynomial x4 - 2x2 - 2 over Q and K be a

splitting field of f(x). Determine the Galois group Gal(K/Q), and find the number of Galois extensions of Q inside K. Prove your answer.

F3. Let f(x) be an irreducible polynomial over the field F and let K / F be a finite extension. (a) Define what it means for the extension K/ F to be normal. (b) Show that if K is normal over F, then, in K[X], f(x) fac

tors into a product of irreducible polynomials of the same degree.

( c) Show by example that this result does not hold for K not normal.

Linear Algebra LAl. Let A be an N by N matrix with entries in C.

(i) Let g be an invertible N by N matrix. Show that limn___,00 An= 0 if and only if limn___,00 (gAg- 1 r = 0.

(ii) Give necessary and sufficient conditions in terms of the conjugacy class of A only for limn___,00 An = 0 to hold.

Here, if An = ( an,(i,j)) is a sequence of N by N matrices with entries an,(i,j), limn___,00 An = 0 if and only if limn___,00 an,(i,j) = 0 for all i and j.

LA2. Let A be a 3 by 3 matrix with complex entries. Suppose that A satisfies the relation A 2 + A+ 13 = 0, where J3

denotes the 3 by 3 identity matrix. (i) List the possible Jordan normal forms of A.

(ii) Suppose A has entries in R. List the possible Jordan normal forms of A.

LA3. Let F be a field of characteristic 0 and let Q be an invertible n by n symmetric matrix with entries in F.

(i) Show that there exists an invertible matrix A such that AQAt is diagonal.

(ii) Does the same remain true if Q is not invertible? Explain.


Test Instructions: Each problem is worth 20 points. You should do two problems in each section. In particular, do not submit more than two problems in each section. In each problem, you must explain clearly your reasoning and any theorems that you quote. Whether you pass or fail depends on your performance in each section, not only on the total score.

GROUPS

PROBLEM Gl Let p be a prime number. Show that a subgroup G of Sp which contains an element of order p and which contains a transposition must be the whole of Sp.

PROBLEM G2 Let G = D2n be the dihedral group of order 2n where n 2". 3. Prove that Aut( G) is isomorphic to the group of 2 x 2 matrices of the form

H = { ( ~ f) : a E (Z/n)*,~ E Z/n }-

PROBLEM G3 Let K be a normal subgroup of a finite group G. Let p be a prime. Let Np be the number of p-Sylow subgroups (p-SSG) of G and N; the number in K.

(a) Show that NP= [G: N0 (P)] where Pis any p-SSG of G and Nc(P) is the normalizer of P in G.

(b) Prove that N; divides NP.

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RINGS

PROBLEM RI Let D be an associative ring with unit having no zero divisors. Assume that the center of D contains a field k such that dimk(D) < oo. Prove that D is a division algebra (i.e., every non-zero element is invertible).

PROBLEM R2 Let G be a finite group of order IGI > 1. The rational group ring Q[G] of G is the Q-algebra consisting of all finite linear combinations

I: agg gEG

where a9 E Q. Multiplication in Q[C] is defined by extending the group multiplication linearly.

(a) Show that Q[C] has a non-trivial idempotent: :3 a E Q[G], a=/= 0, l with a2 = a. Hint: reduce to the case of a cyclic group G = (x: xn = 1).

(b) Show that Q[C] contains an invertible element u that is non-trivial, that is, not of the form u = ag where a E Q and g EC.

(FYI: The Kadison-Kaplansky Conjecture claims that for C without torsion, the group algebra Q[C] contains no non-trivial idempotent. This conjecture is open in general.)

PROBLEM R3 Let R be a Noetherian ring and I any ideal of R. Prove that there exist prime ideals Pi, ... , Pm of R such that

PiP2···Pm c I

Hint: Show that if J is any non-prime ideal, then there exist a, b ¢:. J such that ( J + a) ( J + b) C J. Then use the N oetherian property.

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FIELDS

PROBLEM Fl Consider the polynomial P(X) = X 5

- 4X + 2 in Q[X].

(a) Show that Pis irreducible and has 3 real roots and 2 complex ones.

(b) Show that the Galois group of P is S5 .

PROBLEM F2 Let (n = exp(21ri/n) be a primitive nth root of unity. Let Fn = Q((n)Set

(a) Let n = 6. Find an irreducible polynomial of degree d6 in Q[x] whose roots generate F6 .

(b) Let n = 12. Find an irreducible polynomial of degree d12 in Q[x] whose roots generate F12.

PROBLEM F3 Let E / F be a finite, separable extension of fields. Prove that there exists a EE such that E = F(a). State clearly any theorems you use in the proof.

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LINEAR ALGEBRA

PROBLEM LAI Let V and W be vector spaces over a field F and let V* be the dual space of V. Let Hom(V, W) be the space of linear maps from V to W. There exists a natural linear map

T : V* @F W -----+ Hom(V, W)

defined by T (f @ w) ( v) = f ( v) · w. Show that V is finite dimensional if and only if Tis an isomorphism for all W.

PROBLEM LA2 Let M4 (Q) be the ring of all 4x4 matrices with coefficients in Q. Find a set of representatives for the conjugacy classes of elements X E M4 (Q) satisfying the equation X 4 = 2X 2 .

PROBLEM LA3 Let V be a finite dimensional F-vector space and T : V -----+ V a linear endomorphism. Show that there exists a decomposition

V=ViEBVi

with the properties:

(1) T(¾) C ¾ for i = 1, 2 (2) Tis an isomorphism on Vi (3) T is nilpotent on Vi.

Hint: Consider the sequences of subspaces Im(T) :) Im(T 2) :) • • • and

that Ker(T) c Ker(T 2) c · · · .

I •



Groups

Gl. Let G be a finite group of order g and Z[G] C IQ[G] be the group algebras of G with integer and rational coefficients, respectively. Let

eZ[G] = { ea E IQ[G] I a E Z[G]}

for e = g- 1 'I:hEG h E IQ[G], and define a group

G' = eZ[G]/(Z[G] n eZ[Gl).

Prove that G' is a group of order g. Find a necessary and sufficient condition to have G' ~ G as groups, and justify your answer.

G2. Prove or disprove: (a) the group GL2(1Q) of 2 x 2 matrices with rational coefficients has

finite cyclic subgroups of order bigger than any given positive integer N.

(b) the group GL2(IB.) of 2 x 2 matrices with real coefficients has finite cyclic subgroups of order bigger than any given positive integer N.

G3. Let G be an additive abelian group such that multiplication by n: x f----, nx is surjective for all positive integers n. Let

G[n] = {x E GI nx = o} and p be a prime. (a) Prove that for a given integer m 2: 1, there are only finitely

many subgroups Hof order pm in G if G[p] is finite; (b) Find a formula of the number of subgroups of G of order p if

the order of G[p] is p3, and justify your answer.

Rings

Rl. Let A= Mn(F) be the ring of n x n matrices with entries in a field F. (a) Prove that any left ideal of A is principal of the form Ax; (b) How many left ideals of A if n = 2 and Fis a finite field? (Give

a simple formula of the number of maximal left ideals of A, and justify your answer.)

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R2. Let A be a domain and B = A[T, ,1'] for an indeterminate T. Prove that the ring automorphism group Aut(B;A) of B inducing the identity on A is finite if and only if the group of invertible elements of A is finite.

R3. Consider the covariant functor F : A r-+ Ax from the category ALG of commutative rings with identity to the category of sets. Here Ax is the group of invertible elements of A. Give an explicit form of a commutative ring R such that the functor F is isomorphic to the functor Ar-+ HomALc(R, A).

Fields

Fl. Let L/ F be a cubic (of degree 3) field extension of characteristic zero. Prove that there is an element a E F and a cubic field extension Lo of the field Fo = (Q)( a) such that L is the composite F Lo of F and Lo over Fo.

F2. A field extension L/ Fis said to be balanced if every field homomorphism L ----> L over F is an isomorphism.

(a) Prove that every algebraic (possibly infinite) field extension is balanced;

(b) Give an example of a balanced non-algebraic field extension.

F3. Let p be a prime integer and Fa field such that the degree of every nontrivial finite field extension of Fis divisible by p. Prove that for any finite field extension L/ F, there exists a tower of field extensions F = Fo c F1 c · · · c Fn = L such that [Fi+l : Fi] = p for any i = 0, ... ,n - l.

Linear Algebra

11. Let V be a finite dimensional vector space of dimension n over a field of characteristic 2. A bilinear form b on V is called symmetric (respectively, alternating) if b( v, v') = b( v', v) for all v, v' E V (respectively, b(v, v) = 0 for all v EV). Prove that the space Alt(V) of all alternating bilinear forms on V is a subspace of the space Sym(V) of all symmetric bilinear forms on V and find dimension of the factor space Sym(V)/ Alt(V).

12. Find the number of conjugacy classes of elements of order 4 in the general linear group GL4((Q)).

13. An n x n matrix A over a field F is called regular over F if the minimal and characteristic polynomials of A coincide. Prove that for a field extension L / F, an n x n matrix A over F is regular over F if and only if A is regular over L.

ALGEBRA QUALIFYING EXAM: Spring 2009

TEST INSTRUCTIONS All problems are worth 20 points. You are expected to do 2 problems from each of the four sections. Your total score will be computed by taking the two best-scoring problems in each section. In problems where arguments must be given, you will lose points if you fail to state clearly the basic results that you use.

GROUPS

Gl. Let N be a normal subgroup of a finite group G and P a Sylow p-subgroup of G. Prove that P n N is a Sylow p-subgroup of N.

G2. Let A be an abelian group generated by n elements. Prove that any subgroup of A can be generated by n elements.

G3. Let G be a finite group and H c G a subgroup of index n. Suppose that xH n Hy -=I 0 for any elements x, y E G \ H. Prove that IGI 2: n 2 - n. (Hint: consider an action of G on (G/H) x (G/H).)

RINGS

Rl. Show the the ring Z[2i] consisting of all complex numbers a+ 2bi with a, b E Z is not a PID.

R2. Let Mn(F) be a the matrix ring of n x n matrices over a field F. Suppose that there is a subring of Mn(F) isomorphic to Mm(F) for some m. Prove that m divides n.

R3. Two polynomials f, g E R[t] over a commutative ring Rare called coprime over Riff and g generate the unit ideal in R[t]. Let f,g E Z[t] be two polynomials such that f and g are coprime over Q and the residues off and g in (Z/pZ)[t] are coprime for every prime integer p. Prove that f and g are coprime over Z.

LINEAR ALGEBRA

LAl. A matrix N is said to be nilpotent of order k if Nk = 0, but Nk-I -=I 0. If N is nilpotent of order k, prove that

LA2. Consider the matrix

(A) Prove that (I - A)(I + A)- 1 is an orthogonal matrix. (B) Compute eAt as a function oft.

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LA3. Let 7r E Sn be a permutation of n elements. Let P1r be the n x n matrix taking the standard basis vector ei f---> e1r(i) for all i. Describe the eigenvalues over C of P1r in terms of the cyclic decomposition of 1r.

FIELDS

Fl. Let p(x) E K[x] be a monic irreducible polynomial of degree n whose discriminant D -/= 0, where K has characteristic -/= 2. Prove that the Galois group of p is contained in the alternating group An if and only in vD EK.

F2. Let a be transcendental over Q. What is the minimal polynomial of a over Q( ':,2i/ )?

F3. (A) Which roots of unity are contained in quadratic extensions of Q, and which extensions are these? (B) If K/Q is any field extension that contains a primitive n'th root of unity, and n is odd, then prove K contains a primitive 2n'th root of unity.

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Fall 2009

Test Instructions: Each problem is worth 20 points. Attempt at least 8 problems in any section. All tried problems will be graded.

Part 1: Categories and Functors. ( Cat 1). Let Top be the category of topological spaces. Recall that a morphism f in some

category is called a monomorphism if, for any two morphisms 91 and 92 that can be precomposed with f, f91 = f92 implies 91 = 92- Dually, f is called an epimorphism if, for any 91 and 92 that can be post-composed with f, 9if = 9d implies 91 = 92. (a) Show that a continuous map f: X-> Y is a monomorphism in Top if and only

if f is one-to-one. (b) Now show by example that an epimorphism in Top need not be onto.

(Cat 2). Let F: Ab-> Sets be the forgetful functor from abelian groups to sets. Show that F does not have a right adjoint.

Part 2: Groups. (Gr 1). Suppose A is an abelian group that is generated by n elements (or fewer). Show that

any subgroup of A also can be generated by n elements (or fewer). (Gr 2). Let p < q be primes, n 2'. 0 an integer and G a group of order pqn. Show that G is

solvable.

Part 3: Representations. (Rep 1). Let G be a finite group and p : G -> Gl(V) a complex representation. Prove that

(V, p) splits as a direct sum of irreducible representations of G. [Note: It does not suffice to just quote a theorem. You have to actually prove the statement.]

(Rep 2). Let G be a finite p--group and p: G -> Gl(V) a representation in a IFp-vector space. (a) Show that V has a one-dimensional G-invariant subspace W. (b) Show by example that (V, p) need not split into a direct sum of irreducible

representations.

Part 4: Commutative Rings. (Cl). Find a homomorphism A -> B of commutative rings (sending the identity of A to

the identity of B) and non-zero A-modules M, N such that the canonical map

B 0A HomA(M, N) -> Homs(B '61A M, B '61A N)

is the zero map, and justify your answer. Prove that the map is an isomorphism if M is a finitely generated projective A-module.

( C2). Prove the following facts: (a) Any subring of IQ! sharing the identity with IQ! is a PID. (b) For a subring A c Z[A] sharing the identity with Z[A], if A =f-Z and

A =f-Z[A], A is not a PID. 1

2 Part 5: Non-commutative Rings. (Rl). Prove that every two-sided ideal of the ring M2(Z) is principal, i.e., generated by one

element. (R2). Let B be a central simple algebra over k of dimension 4 (so, the center of Bis k and

has no nontrivial two-sided ideals except for (0) and B itself). Prove the following facts. (a) All left ideals of B have even dimensional. (b) B ~ M2 (k) if and only if Bis not a division algebra, where M2 (k) is the matrix

algebra of 2 x 2 matrices with coefficients in k.

Part 6: Fields. (Fl). Prove that the multiplicative group F \ {O} of a field Fis a cyclic group if and only

if F is a finite field. (F2). Let k = IF2 (t, s) be the field of fractions of two variable polynomial ring IF 2[t, s], where

IF2 is the field with 2 elements. Write 0a for a root of T 2 + T +a= 0 for a E kin an algebraic closure of k. An intermediate field M between K and k for a field extension K / k is a subfield in K containing k. (a) How many intermediate fields between k and k(01, 08 )? (b) How many intermediate fields between k and k( ,Jt, ,js)?

Justify all your answers.


Test Instructions: Each problem is worth 20 points. Attempt at least 8 problems. All tried problems will be graded.

Part 1: Categories and Functors

Problem 1 Show that the functor from (unitary) rings to groups sending a ring A to its group of units Ax is co-representable by a ring R. In other words,· show there exists a ring R and a natural isomorphism of functors Homrings(R,A)-+ Ax.

Problem 2 (i) Define what is means for two categories to be equivalent. (ii) A groupoid g is a category such that all morphisms are isomorphisms. g is called connected if for any two objects x and y, Homg(x,y) is non-empty. Show that any non-empty connected groupoid is equivalent to a group, that is, a groupoid with one object.

Part 2: Groups

Problem 3 Determine, using the structure theory of abelian groups or otherwise, all finitely generated abelian groups A such that the group Aut(A) of automorphisms of A is finite. State clearly any basic theorems that you use. Determine the order of the automorphism group of A= Z x Z/4Z.

Problem 4 Show that if S4 denotes the symmetric group of degree 4, and a is an automorphism of S4 , and r E S4 is a transposition, then a(r) is also a transposition. By studying the action of a on transpositions, show that every automorphism of S4 is inner. (Remark: this result holds for all Sn except n=6.)

Part 3: Representations

Problem 5 Give the total number and the dimensions of the irreducible complex representations of S4 . Prove your answer.

Problem 6 State and prove Schur's Lemma which describes Homc(V, W) for V and W finite dimensional irreducible complex representations of a finite group.

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Part 4: Commutative Rings and Modules

Problem 7 Show that the group of units of the ring Z/NZ is cyclic iff N is either a power of an odd prime number, twice a power of an odd prime number, or 4.

Problem 8 Let F be the field with 2 elements and let R = F[X]. List up to isomorphism all R-modules with 8 elements that are cyclic.

Part 5: Non-Commutative Rings

Problem 9 Let A be a left Noetherian ring. Show that every left invertible element a E A is two-sided invertible.

Problem 10 Let F be a field and V a finite-dimensional F-vector space. Show that R = EndF(V) has no non-trivial two-sided ideals.

Part 6: Fields

Problem 11 Let F be a field of characteristic zero containing the p-th roots. of unity for p a prime. Show that the cyclic extensions of degree p of F in any algebraic closure F of F are in one to one correspondence with the subgroups of order p of F* /(F')P .

Problem 12 Determine all fields F such that the multiplicative group of F is finitely generated.

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Test Instructions: Each problem is worth 20 points. Attempt at least 8 problems. All tried problems will be graded.

Part 1: Categories and Functors

Problem 1 Let Grp be the category of groups and Ab the category of abelian groups. If F: Ab -+ Grp is the inclusion of categories, then find a left adjoint to F and prove that it is a left adjoint.

Problem 2 Let F: C -+ Sets be a covariant functor on a. category C. Assume that Fis representable by a.n object CF E Obj(C). Identify which of the following statements are necessarily correct, proving your answers in each case. (i) If CE Obj(C) and F(C)-/- 0, then there is an element f E Morc(CF,C); (ii) If G is a. left adjoint of F, then G is representable; (iii) If C, D E Obj(C), and there is a map of sets .f: F(C) -+ F(D), then there exists a g E Morc(C, D). (iv) If C,D E Obj(C), and there exists a g E Morc(C,D), then there exists a map of sets .f: F( C) -+ F(D) (note we a.re not guaranteed in advance that F(D) -/-0). (v) If h E Morc(D, CF), then for any C E Obj(C), there is a map of sets F(C)-+ Morc(D,C).

Part 2: Groups

Problem 3 Prove that there is no simple group of order 120.

Problem 4 1. Show from first principles (i.e. without using the classifi-cation theorem) that a. subgroup of a finitely generated abelia.n group is finitely generated.

2. Let M <;;; Z3 be the subgroup generated by elements (13, 9, 2), (29, 21, 5), and (2, 2, 2). Determine the isomorphism class of the quotient group Z3/M.

Part 3: Representations

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Problem 5 Prove that if a finite group G acts transitively on a set S having more than one element then there exists an element of G which fixes no element of S. (Hint: first prove a formula that counts the average number of fixpoints of an arbitrary (i.e. not necessarily transitive) action.)

Problem 6 Let R be the 5 dimensional tautological representation of S5 .

Show that R is isomorphic to the direct sum of the trivial representation and an irreducible 4 dimensional representation. State clearly any principles or theorems that you use.

Part 4: Commutative Rings and Modules

Problem 7 Let k be a field and let f be an irreducible element of the polynomial ring k[x, y]. (i) Describe the localization k[x, y]p where P = (!). Prove that it is a subring of the field of rational functions k(x, y). (ii) For any r E k(x, y), prove that

r E Im(k[x,y]p--+ k(x,y))

for all but finitely many choices of f.

Problem 8 Let k be an algebraically closed field, R = k[x1 , ... , Xn] and f: R--+ Rd be given by f(p) = (pf 1 , ... ,Pfd) with all fi ER. Let M be the ideal generated by x 1 - a1, ... , Xn - an, where ai E k for all i. Consider for an integer r, r :::0: 1, the map

induced by f. Prove that ker(f M') # 0 if and only if fj ( a1, ... , an) ~ 0 for all j.

Part 5: Non-Commutative Rings

Problem 9 Let R be a ring with 1. Show that the only two-sided ideals of Mnxn(R), then x n matrices with entries in R, are of the form Mnxn(I) for some 2-sided ideal I in R.

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Problem 10 Let G be a group and Zc the integral group ring, i.e. the set of formal finite sums ~i ni(9i) where ni E Zand 9i E G for all i. Multiplication f · h of two elements f = ~i ni(9i) and h = ~j m;(9j) is

f · h = L nimj(9i9j), i,j

Let J be the two-sided ideal

I= {L ni(9i) I L ni = O}. i

Construct a natural map

F:J/J 2 ➔ G/[G,G]

and prove that this map is an isomorphism of Z-modules.

Part 6: Fields

Problem 11 Show that the extension of Q generated by 51/ 2 + 21/ 3 is equal to Q[51/2, 31/3].

Problem 12 Show that the multiplicative group of a finite field is cyclic and use this result to prove that the polynomial X 4 + 1 is never irreducible over any finite field.

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1. Let G be a group of order n. Show that there are two subgroups H 1 and H 2

of the symmetric group Sn, both isomorphic to G such that h1h2 = h2h1 for all h1 E H1 and h2 EH,.

2. Let G be a finitely generated group. Shmv that G contains only finitely many subgroups of any fixed finite index.

3. Let R be a noetherian domain. Show that every nonzero nonunit in R is a product of irreducible elements and that Risa UFD if and only if every nonzero nonunit in R is a product of prime elements, where an element is prime if it generates a nonzero prime ideal.

4. Prove that every prime ideal in Z[l] can be generated by two elements.

5. Let F be a field having no nontrivial field extensions of odd degree and K / F a finite field extension. Show if K has no field extensions of degree two) then F is perfect and K is algebraically closed.

6. Prove that over a finite field there are irreducible polynomials of any positive degree.

7. Let T : V ------+ V be a linear operator on a finite dimensional vector space. Prove that the characteristic polynomial Pr is irreducible if and only if T has no nontrivial invariant subspaces.

8. Let V be a finite dimensional vector space over a field F. Prove that every right ideal in Endp(V) is of the form {T : im(T) c W} for a unique subspace W of V.

9. Let G be a finite group of invertible linear operators on a finite dimensional vector space V over the field of complex numbers. Prove that if ( dim V) 2 > [GI, then there is a proper non-zero subspace W c V such that g(w) E W for every g E G and every w E W.

10. Show that there is a (covariant) functor from the category of groups to the category of sets taking a group G to the set of all subgroups of G. Determine whether this functor is representable.

ALGEBRA QUALIFYING EXAM FALL 2011

Do all the following 10 problems (see reverse). Good luck!

Problem 1. For a finite field lF, prove that the order of the group SL2 (lF), of 2 x 2 matrices with determinant 1, is divisible by 6.

Problem 2. Let G be a non-trivial finite group and p a prime. If every subgroup H f G has index divisible by p, prove that the center of G has order divisible by p.

Problem 3. Let R be a local UFD of Krull dimension 2 (meaning that the maximal integer m for which there exist strict inclusions of prime ideals Po c;; p1 c;; · · · c;; Pm in R is exactly 2). Let 1r E R be neither zero nor a unit. Prove that R[¼J is a PID.

Problem 4. Let p be a prime. Prove that the nilradical of the ring lFp[X] ®wp[XPJ lFp[X] is a principal ideal.

Problem 5. Let lF be a finite field and lF be an algebraic closure of lF. Let K be a subfield of lF generated by all roots of unity over lF. Show that any simple K-algebra of finite dimension over K is isomorphic to the matrix algebra Mn(K) for a positive integer n.

Problem 6. Let R be a commutative ring and let M be a finitely generated R-module. Let f : M ➔ M be R-linear such that f ® id : M ®RR[T] ➔ M ®RR[T] is surjective. Prove that f is an isomorphism.

Problem 7. Let C be the category of semi-symplectic topological quantum paramonoids of Rice-Paddy type, satisfying the MussoliniRostropovich equations at infinity. Let X, Y be objects of C such that the functors Morc(X, -) and Morc(Y, -) are isomorphic, as covariant functors from C to sets. Show that X and Y are isomorphic in C.

Problem 8. Let r be the Galois group of the polynomial X 5 -9X +3 over l(Ji. Determiner. [Hint: Show that r contains an element of order 5 and that r contains a transposition, in a sense to be made precise.]

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2 ALGEBRA QUALIFYING EXAM FALL 2011

Problem 9. (We denote by IFG the group algebra of G.) (a) Is there a group G with CG isomorphic to C x C x M2 (C)?

(b) Is there a group G with IJJ, G isomorphic to IJJ, x IJJ, x M3 ( !JJ,) ?

Problem 10. Let K/k be an extension of finite fields. Show that the norm N K/k : K ➔ k is surjective.

Algebra Qualifying Exam, Spring 2012

1. Let V be a finite dimensional space over Q and G ⇢ GL(V ) a finite sub-group. Prove that the Q-subalgebra of End(V ) generated by G is semisimple.

2. Let V be the vector space of all a 2 Rn such that a1 + a2 + · · · + an = 0.The symmetric group Sn acts naturally on V . Prove that the Sn-module V issimple.

3. Let R be a commutative local ring, and P a finitely generated projectiveR-module. Show that P is a free module.

4. Let R be the subring of M3(R) consisting of all matrices (aij) with a31 =a32 = 0. Determine the Jacobson radical of R.

5. Let G be a finite group, K a normal subgroup and P a p-Sylow subgroupof G. Prove that P \K is a p-Sylow subgroup of K.

6. Determine the Galois group of the polynomial X8 + 16 over Q.

7. Let F be a finite field with pn elements (p is prime). Find the number ofelements of F that can be written in the form ap � a for some a 2 F .

8. Let R be a reduced (meaning, no non-zero nilpotent elements) commutativering that has a unique proper prime ideal. Show that R is a field.

9. Let R be a flat commutative Z-algebra, and ModR the category of R-modules. Suppose that I is an injective R-module. Show that the underlyingabelian group of I is divisible.

10. Determine the automorphism group of the symmetric group S3 up toisomorphism.

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Algebra Qualifying Exam, Fall 2012

(1) Let p be a prime integer and let G be a (finite) p-group. Write C for the subgroup of central elements x E G satisfying xP = 1. Let N be a normal subgroup of G such that N n C = {1}. Prove that N = {1}.

( 2) Let A be an n x n matrix over a field F having only one invariant factor. Prove that every n x n matrix over F that commutes with A is a polynomial in A with coefficients in F.

(3) Let F be a field and let n be a positive integer such that F has no nontrivial field extensions of degree less than n. Let L = F( x) be a field extension with xn E F. Prove that every element in L is a product of elements of the form ax+ b with a, b E F.

( 4) Let F be the functor from the category of rings to the category of sets taking a ring R to the set { x2 I x E R}. Determine whether F is representable.

(5) Let G and H be finite groups and let V and W be irreducible ( over C) Gand H-modules respectively. Prove that the G x H-module V ® W is also irreducible.

( 6) Let Dn be a dihedral group of order 2n > 4; so, it contains a cyclic subgroup C of order n on which u E Dn outside C acts as ucu- 1 = c 1 for all c E C. When is the cyclic subgroup C with the above property unique? Determine all n for which Dn has unique cyclic subgroup C and justify your answer.

(7) Let D be a central simple division algebra of dimension 4 over a field F. If a quadratic extension K/ F can be isomorphically embedded into D as Falgebras, prove that D®pK is isomorphic to the 2 x 2 matrix algebra M2 (K) as K-algebras.

(8) How many manic irreducible polynomials over IFP of prime degree l are there? Here IFP is the field of p elements for a prime number p. Justify your answer.

(9) Consider a covariant functor F : R f-+ Rx from the category of commutative rings with a multiplicative identity into the category of sets. Let G = Autfunctors(F) be made up of natural transformations I: F--+ F having an inverse J : F --+ F E G such that I o J = J o I is the identity natural transformation. Prove first that F is representable by a ring, that G is a finite set, and find the order of the group G with justification.

(10) For a finite field IF of order q, consider the polynomial ring R = IF[x], and let L be a free R-module of rank 2. Give the number of R-submodules M such that xL c;;; M c;;; L, and justify your answer.

1


1: Let G be a free abelian group of rank r, so G is isomorphic to Zras groups. Show that

G has only finitely many subgroups of a given finite index n.

2: Assume that L is a Galois extension of the field of rational numbers Q and that K ⇢ Lis the subfield generated by all roots of unity in L. Suppose that L = Q[a], where an 2 Qfor some positive integer n. Show that the Galois group Gal(L/K) is cyclic.

3: Let K ⇢ L be an algebraic extension of fields. An element a of L is called abelian if K[a]is a Galois extension of K with abelian Galois group Gal(K[a]/K). Show that the set of

abelian elements of L is a subfield of L containing K.

4: Let F2 be the field with 2 elements and let R = F2[x]. List, up to isomorphism, all

R–modules with 8 elements.

5: Let R be a commutative local ring, so R has a unique maximal ideal M.a) Show that if x 2 M then 1� x is invertible.

b) Show that if R is Noetherian and if I is an ideal such that I2 = I then I = 0.

6: Let D be a division ring of characteristic 0. Assume that D has dimension 2 as a Q-vector

space. Show that D is commutative.

7: Let F = F2 be the field with two elements. Show that there is a ring homomorphism

F [GL2(F )] ! M2(F ) that sends the element g in the group ring to the matrix g 2 M2(F ).

Show that this homomorphism is surjective. Let K be the kernel; since it is a left ideal,

it is a (left) GL2(F )-module. Is this module indecomposable? (Reminder: a module is

indecomposable if it is not the direct sum of two proper submodules.) Describe the simple

modules in its composition series.

8: Let C and D be additive categories, and let � : C ! D be a functor. Show: If � has a

right adjoint, then � commutes with direct sums and for any two objects x and y in C, the

map �x,y : HomC(x, y) ! HomD(x, y) is a homomorphism.

9: Let D be an associative ring without zero divisors, and assume the center of D is a field

over which D is a finite-dimensional vector space. Prove that D is a division algebra.

10: Let G be a finite group of order n and ⇢ : G ! GL(V ) a complex representation of Gof dimension n. Show that ⇢ cannot be irreducible.


Do the following ten problems.

1. How many groups are there up to isomorphism of order pq where p > qare prime integers?

2. Show that there are up to isomorphism exactly two nonabelian groups Gof order 8. Prove that each of them has an irreducible complex representationof dimension 2.

3. For a positive integer n, let Φn(X) be the nth cyclotomic polynomial. Ifa is an integer and p a prime not dividing n, such that p divides Φn(a), showthat the order of a mod p is n. Using this prove that there are infinitelymany primes p such that p is 1 modulo n.

4. Given a field K of characteristic p, when is an α that is algebraic over Ksaid to be separable? Show that if α is algebraic over K, then α is separableif and only if K(α) = K(αpn) for all positive integers n.

5. Let G be a finite group which has the property that for any element g ∈ Gof order n, and an integer r prime to n, the elements g and gr lie in the sameconjugacy class. Then show that the character of every representation of Gtakes values in the rational numbers Q (in fact even the integers Z). (Hint:Use Galois theory.)

6. Let I be an ideal of a commutative ring and a ∈ R. Suppose the idealsI + Ra and (I : a) := {x ∈ R | ax ∈ I} are finitely generated. Prove that Iis also finitely generated.

7. Give an example of a 10 × 10 matrix over R with minimal polynomial(X + 1)2(X4 + 1) which is not similar to a matrix with rational coefficients.

8. Suppose that E/F is an algebraic extension of fields such that everynonconstant polynomial in F [X] has at least one root in E. Show that E isalgebraically closed.

9. Prove that is ab = 1 in a semisimple ring, then ba = 1.

1

10. Let A be the functor from the category of groups to the category of(unital) rings taking a group G to the group ring Z[G] of all finite formalsums

!g∈G agg with ag ∈ Z (the product in Z[G] is induced by the group

operation in G). Prove that A has a right adjoint functor.

2


Please do the following ten problems. Write your UID number ONLY, not your name.

( 1.) Let L : C --+ D be a functor, left adjoint to R : D ➔ C. Show: if the counit Lo R ➔ idD is a natural isomorphism, then R is fully faithful.

(2.) Let A be a central division algebra (of finite dimension) over a field k. Let [A, A] be the k-subspace of A spanned by the elements ab - ba with a, b E A. Show that [A, A] cf A.

(3.) Given ¢ : A ➔ B a surjective morphism of rings, show that the image by ¢ of the Jacobson radical of A is contained in the Jacobson radical of B.

( 4.) Let G be a group and H a normal subgroup of G. Let k be a field and let V be an irreducible representation of G over k. Show that the restriction of V to H is semisimple.

(5.) Let G be a finite group acting transitively on a finite set X. Let x EX and let P be a Sylow p-subgroup of the stabilizer of x in G. Show that Na(P) acts transitively on XP_

(6.) Let A be a ring and Ma noetherian A-module. Show that any surjective morphism of A-modules M ➔ M is an isomorphism.

(7.) Let G be a finite group and lets, t E G be two distinct elements of order 2. Show that the subgroup of G generated by s and t is a dihedral group. (Recall that the dihedral groups are the groups D(m) = (g, hlg2 = h2 = (gh)m = 1) for some m 2: 2).

(8.) Let F be a finite field. Without using any of the theorems on finite fields, show that F has a field extension of degree 2.

(9.) Let G be a finite group. Show that there exist fields F C E such that E / F is Galois with group G.

(10.) Let F be a field. Show that the polynomial ring F[t] has infinitely many prime ideals. Also prove that algebraically closed fields are infinite.


Please do the following ten problems. Write your UID number ONLY, not your

name.

(1) Let G be a finite group. Let Z[G] be the group algebra of G with augmentation ideal a.Show that a/a2 ⇠= G/G

0 as abelian groups for the derived group G0 of G.

(2) Let Fp denote the finite field of p elements. Consider the covariant functor F from thecategory of commutative Fp-algebras with a multiplicative identity to abelian groups sendinga ring R to its p-th roots of unity, that is, F (R) = {⇣ 2 R|⇣p = 1}. Answer the followingquestions and justify your answers.

(a) Give an example of a finite local ring R such that F (R) has p2 elements.(b) Let Aut(F ) be the set of natural transformations of F into itself inducing a group

automorphism of F (A) for all commutative rings A with identity. Prove that F isrepresentable and use the Yoneda Lemma to compute the order of Aut(F ).

(3) Pick a non-zero rational number x. Determine all possibilities for the Galois group G ofthe normal closure of Q[ 4

px] over Q, where 4

px is the root of X4 � x with maximal degree over

Q.

(4) Let D be a 9-dimensional central division algebra over Q and K ⇢ D be a field extensionof Q of degree > 1. Show that K ⌦Q K is not a field and deduce that D ⌦Q K is no longer adivision algebra.

(5) Let R be a commutative algebra over Q of finite dimension n. Let ⇢ : R ! Mn(Q) bethe regular representation, and define Tr : R ! Q by the matrix trace of ⇢. If the pairing(x, y) = Tr(xy) is non-degenerate on R, prove that R is semi-simple.

(6) Let G be a finite group and let p be the smallest prime number dividing the order of G.Assume G has a normal subgroup H of order p. Show that H is contained in the center of G.

(7) Let G be a finite group and P a Sylow 2-subgroup of G. Assume P is cyclic, generatedby an element x. Show that the signature of the permutation of G given by g 7! xg is �1.Deduce that G has a non-trivial quotient of order 2.

(8) Let A be a ring. Assume there is an infinite chain of left ideals I0 ⇢ I1 ⇢ · · · ⇢ A withIi 6= Ii+1 for i � 0. Show that A has a left ideal that is not finitely generated as a left A-module.

(9) Let A be a ring and let i, j 2 A such that i2 = i and j2 = j. Show that the left A-modules

Ai and Aj are isomorphic if and only if there are a, b 2 A such that i = ab and j = ba.

(10) Let n be a positive integer. Let An be the Q-algebra generated by elements x1, . . . , xn,y1, . . . , yn with relations

xixj = xjxi, yiyj = yjyi and yixj � xjyi = �ij for 1 i, j n.

Show that there is a representation of An on the vector space Q[t1, . . . , tn] where xi acts bymultiplication by ti and yi acts as

@@ti

.

1



name.

(1) What are the coproducts in the category of groups?

(2) Let C be the category of groups and C 0 be its full subcategory with objects the abeliangroups. Let F : C 0 ! C be the inclusion functor. Determine the left adjoint of F and showthat F has no right adjoint.

(3) Let R be a ring. Show that R is a division ring if and only if all R-modules are free.

(4) Let M = Z[1p ]/Z and N = Q/Z, where Z[1p ] ⇢ Q is the subring generated by 1p for a

prime p. Show that

(a) M is an artinian module but not a noetherian module;(b) N is neither noetherian nor artinian.

(5) Let K and L be quadratic field extensions of a field k. Prove that K ⌦k L is an integraldomain if and only if the k-algebras K and L are not isomorphic.

(6) Let K ⇢ L be subfields of C and let p be a prime. Assume K contains a non-trivial p-throot of unity. Show that L/K is a degree p Galois extension if and only if there is an elementa 2 K that does not admit a p-th root, such that L = K( p

pa).

(7) Determine the ring endomorphisms of F2[t, t�1], where t is an indeterminate.

(8) Let G be a finite group of order pq, where p and q are distinct primes. Show that

(a) G has a normal subgroup distinct from 1 and G(b) if p 6⌘1 (mod q) and q 6⌘1 (mod p), then G is abelian.

(9) Let G be a finite group of order pn for a prime p. Show that the group ring Fp[G] overthe finite field Fp with p elements has a unique maximal two-sided ideal.

(10) Let E, M and F be finite abelian groups and consider group homomorphisms Ef�!

Mg�! F . Assume g is injective. Show that |Coker(g � f)| = |Coker(g)| · |Coker(f)| where |X|

denotes the order of a finite set X.

1


Please do the following ten problems. Write your UID number ONLY, not your name.

(1) Show that the inclusion map Z Y «Jl is an epimorphism in the category of rings with multiplicative identity.

(2) Let R be a principal ideal domain with field of fractions K.

(a) Let S be a non-empty multiplicatively closed subset of R \ {0}. Show that R[s- 1] is a principal ideal domain.

(b) Show that any subring of K containing R is of the form R[ s- 1 J

for some multiplicatively closed subset S of R \ {0}.

(3) Let k be a field and define A= k[X, Y]/(X 2 ,XY, Y 2).

(a) What are the principal ideals of A? (b) What are the ideals of A?

(4) Let K be a field and let L be the field K(X) of rational functions over K.

(a) Show that there are two unique K-automorphisms f and g of the field L = K(X) such that f(X) = x- 1 and g(X) = 1 - X. Let G be the subgroup of the group of K-automorphisms of L generated by f and g. Show that IGI > 3.

(b) Let E = L 0 . Show that

(X 2 - X + 1)3

p = X2(X - 1)2 E E.

(c) Show that L/K(P) is a finite extension of degree 6. (d) Deduce that E = K(P) and that G is isomorphic to the sym

metric group S3 .

(5) (a) Let G be a group of order pev with v and e positive integers,

JJ prime, JJ > v, and v not a multiple of p. Show that G has a normal Sylow p-subgroup.

(b) Show that a nontrivial finite p-group has nontrivial center.

1

2

( 6) Let F be a field of characteristic not 2. Let a and b be nonzero elements of F. Let R be the F-algebra

R = F(i,j)/('i 2 - a,j2-b,ij + ji),

the quotient of the free associative algebra on 2 generators by the given two-sided ideal.

(a) Let F be an algebraic closure of F. Show that R 0F F is isomorphic as an F-algebra to the matrix algebra lvh(F).

(b) Give a basis for Ras an F-vector space, justifying your answer. (You may use (a).)

(7) Show that the symmetric group S4 has exactly two isomorphism classes of irreducible complex representations of dimension 3. Compute the characters of these two representations.

(8) Let F be a field. Show that the group SL(2, F) is generated by

the matrices ( t 1) and ( ! ~) for elements e in F.

(9) ( a) Let R be a finite-dimensional associative algebra over a field

F. Show that every element of the Jacobson radical of R is nilpotent.

(b) Let R be a ring. Is an element of the Jacobson radical of R always nilpotent? Is a nilpotent element of R always in the Jacobson radical? Justify your answers.

(10) Let p be a prime number. For each abelian group K of order p2 , how many subgroups H of Z3 are there with Z3 / H isomorphic to K?

Algebra Qualifying Exam, Spring2016

Please do the following ten problems. Write your UID num-

ber ONLY, not your name.

(1)(a) Give an example of a unique factorization domain A that is not

a PID. You need not show that A is a UFD (assuming it is),

but please show that your example is not a PID.

(b) Let R be a UFD. Let p be a prime ideal such that 0 6= p and

there is no prime ideal strictly between 0 and p. Show that p is

principal.

(2) Consider the functor F from commutative rings to abelian groups

that takes a commutative ring R to the group R⇤of invertible elements.

Does F have a left adjoint? Does F have a right adjoint? Justify your

answers.

(3) LetR be a ring which is left artinian (that is, artinian with respect

to left ideals). Suppose that R is a domain, meaning that 1 6= 0 in Rand ab = 0 implies a = 0 or b = 0 in R. Show that R is a division ring.

(4) Let A be a commutative ring, S a multiplicatively closed subset

of A, A ! A[S�1] the localization.

(a) Which elements of A map to zero in A[S�1]?

(b) Let p be a prime ideal in A. Show that the ideal generated by

the image of p in A[S�1] is prime if and only if the intersection

of p with S is empty.

(5) Let A be the ring Chu, vi/(uv� vu� 1), the quotient of the free

associative algebra on two generators by the given two-sided ideal.

(a) Show that every nonzero A-module M has infinite dimension

as a complex vector space.

(b) Let M be an A-module with a nonzero element y such that

uy = 0. Show that the elements y, vy, v2y, . . . are C-linearlyindependent in M .

(6) Let K be a field of characteristic p > 0. For an element a 2 K,

show that the polynomial P (X) = Xp � X + a is irreducible over K1

2

if and only if it has no root in K. Show also that, if P is irreducible,

then any root of it generates a cyclic extension of K of degree p.

(7) Show that for every positive integer n, there exists a cyclic ex-

tension of Q of degree n which is contained in R.

(8) Determine the character table of S4, the symmetric group on 4

letters. Justify your answer.

(9) Show that if G is a finite group acting transitively on a set Xwith at least two elements, then there exists g 2 G which fixes no point

of X.

(10)(a) Determine the Galois group of the polynomial X4 � 2 over Q,

as a subgroup of a permutation group. Also, give generators

and relations for this group.

(b) Determine the Galois group of the polynomial X3 � 3X � 1

over Q. (Hint: for polynomials of the form X3+ aX + b, the

quantity � = �4a3 � 27b2, known as the discriminant, plays a

key theoretical role.) Explain your answer.


Please do the following ten problems. Write your UID number ONLY, notyour name.

(1) Let G be a group generated by a and b with only relation a2 = b2 = 1 for thegroup identity 1. Determine the group structure of G and justify your answer.

(2) Let K be a semi-simple quadratic extension over Q and consider the regularrepresentation ρ : K → M2(Q). Compute the index of ρ(K×) in the normal-izer of ρ(K×) in GL2(Q), and justify your answer.

(3) Let A be an integral domain with field of fractions F . For an A-ideal a, provethat a is an A-projective ideal finitely generated over A if there exists anA-submodule b of F such that ab = A, where ab is an A-submodule of Fgenerated by ab for all a ∈ a and b ∈ b.

(4) Let D be a dihedral group of order 2p with normal cyclic subgroup C oforder p for an odd prime p. Find the number of n-dimensional irreduciblerepresentations of D (up to isomorphisms) over C for each n, and justify youranswer.

(5) Let f ∈ F [X] be an irreducible separable polynomial of prime degree over afield F and let K/F be a splitting field of f . Prove that there is an elementin the Galois group of K/F permuting cyclically all roots of f in K.

(6) Let F be a field of characteristic p > 0. Prove that for every a ∈ F , thepolynomial xp − a is either irreducible or split into a product of linear factors.

(7) Let f ∈ Q[X] and ξ ∈ C a root of unity. Show that f(ξ) = 21

4 .

(8) Prove that if a functor F : C → Sets has a left adjoint functor, then F isrepresentable.

(9) Let F be a field and a ∈ F . Prove that the functor from the category ofcommutative F -algebras to Sets taking an algebra R to the set of invertibleelements of the ring R[X]/(X2 − a) is representable.

(10) Let F be a field and A a simple subalgebra of a finite dimensional F -algebraB. Prove that dimF (A) divides dimF (B).

1

Algebra Qualifying Exam, Spring2017

Please do the following ten problems. Write your UID num-ber ONLY, not your name.

(1) Choose a representative for every conjugacy class in the groupGL(2,R). Justify your answer.

(2) Let G be the group with presentation

⟨x, y : x4 = 1, y5 = 1, xyx−1 = y2⟩,

which has order 20. Find the character table of G.

(3) Find the number of subgroups of index 3 in the free group F2 =⟨u, v⟩ on two generators. Justify your answer.

(4) Show that the ring R = C[x, y]/(y2 − x3 + 1) is a Dedekinddomain. (Hint: compare R with the subring C[x].)

(5) Let S be a multiplicatively closed subset of a commutative ring R.For a prime ideal I in R with I ∩ S = ∅, show that the ideal I ·R[S−1]in the localized ring R[S−1] is prime. Also, show that sending I toI · R[S−1] gives a bijection between the prime ideals in R that do notmeet S and the prime ideals in the localized ring R[S−1].

(6) Prove the following generalization of Nakayama’s Lemma to non-commutative rings. Let R be a ring with 1 (not necessarily commuta-tive) and suppose that J ⊂ R is a left ideal contained in every maximalleft ideal of R. If M is a finitely generated left R-module such thatJM = M , prove that M = 0.

(7) Find [K : Q] where K is a splitting field of X6− 4X3+1 over Q.

(8) Let M be an abelian group (written additively). Prove that thereis a functor F from the opposite of the category of rings to the categoryof sets taking a ring R to the set of all left R-module structures on M .Is the functor F representable?

1

2

(9) Let R be a ring. Prove that if the left free R-modules Rn andRm are isomorphic for some positive integers n and m, then Rn andRm are isomorphic as right R-modules.

(10) LetK/F be a (finite) Galois field extension with G = Gal(K/F )and let H ⊂ G be a subgroup. Determine in terms of H and G thegroup Gal(KH/F ) of all field automorphisms of KH over F .



name.

(1) Let G be a finite group, p a prime number, and S a Sylow p-subgroup of G. Let

N = {g 2 G|gSg�1= S}. Let X and Y be two subsets of Z(S) (the center of S) such that

there is g 2 G with gXg�1= Y .

Show that there exists n 2 N such that nxn�1= gxg�1

for all x 2 X.

(2) Let G be a finite group of order a power of a prime number p. Let �(G) be the subgroup

of G generated by elements of the form gp for g 2 G and ghg�1h�1for g, h 2 G.

Show that �(G) is the intersection of the maximal proper subgroups of G.

(3) Let k be a field and A a finite-dimensional k-algebra. Denote by J(A) the Jacobson

radical of A.Let t : A ! k be a morphism of k-vector spaces such that t(ab) = t(ba) for all a, b 2 A.

Assume ker(t) contains no non-zero left ideal. Let M be the set of elements a in A such that

t(xa) = 0 for all x 2 J(A).Show that M is the largest semi-simple left A-submodule of A.

(4) Let R be a commutative noetherian ring and A a finitely generated R-algebra (not

necessarily commutative). Let B be an R-subalgebra of the center Z(A). Assume A is a

finitely generated B-module. Show that B is a finitely generated R-algebra.

(5) Let A be a ring and M an A-module that is a finite direct sum of simple A-modules. Let

f 2 EndZ(M). Assume f � g = g � f for all g 2 EndA(M). Consider a positive integer n.

(a). Show that the map fn : Mn ! Mndefined by fn(m1, . . . ,mn) = (f(m1), . . . , f(mn))

commutes with all elements of EndA(Mn).

(b). Deduce that given any family (m1, . . . ,mn) 2 Mn, there exists a 2 A such that

(f(m1), . . . , f(mn)) = (am1, . . . , amn).

(6) Let R be an integral domain, and let M be an R-module. Prove that M is R-torsion-free

if and only if the localization Mp is Rp-torsion-free for all prime ideals p of R.

(7)

(a). Show that there is at most one extension F (↵) of a field F such that ↵4 2 F , ↵2 /2 F ,

and F (↵) = F (↵2).

(b). Find the isomorphism class of the Galois group of the splitting field of x4 � a for

a 2 Q with a /2 ±Q2.

1

2

(8) Let F be a field, and let f, g 2 F [x] � {0} be relatively prime and not both constant.

Show that F (x) has finite degree d = max(deg(f), deg(g)) over its subfield F (fg ). (Hint: If the

degree were less than d, show that there exists a nonzero polynomial with coe�cients in F [x]of degree less than d having

fg as a root.)

(9) Let R be a commutative ring, and let A, B, and C be R-modules. Suppose that A is

finitely presented over R and C is flat over R. Show that

HomR(A,B ⌦R C) ⇠= HomR(A,B)⌦R C.

(10) Let C be a category with finite products, and let C2be the category of pairs of objects

of C together with morphisms (A,A0) ! (B,B0

) of pairs consisting of pairs (A ! B,A0 ! B0)

of morphisms in C. Let F : C2 ! C be the direct product functor (that takes pairs of objects

and morphisms to their products).

a. Find a left adjoint to F .

b. For C the category of abelian groups, determine whether or not F has a right adjoint.


Please do the following ten problems. Write your UID num-

ber ONLY, not your name.

(1) Let ↵ 2 C. Suppose that [Q(↵) : Q] is finite and prime to n!for an integer n > 1. Show that Q(↵n

) = Q(↵).

(2) Let ⇣9 = 1 and ⇣3 6= 1 with ⇣ 2 C.(a) Show that

3p3 62 Q(⇣),

(b) If ↵3= 3, show that ↵ is not a cube in Q(⇣,↵).

(3) Let Zn(n > 1) be made of column vectors with integer coef-

ficients. Prove that for every non-zero left ideal I of Mn(Z),IZn

(the subgroup generated by products ↵v with ↵ 2 I and

v 2 Zn) has finite index in Zn

.

(4) Let p be a prime number, and let D be a central simple division

algebra of dimension p2 over a field k. Pick ↵ 2 D not in the

center and write K for the subfield of D generated by ↵. Provethat D ⌦k K ⇠= Mp(K) (the p ⇥ p matrix algebra with entries

in K).

(5) Let C be a category. A morphism f : A ! B in C is called

an epimorphism if for any two morphisms g, h : B ! X in C,

g � f = h � f implies g = h. Let ALG be the category of

Z-algebras, and let MOD be the category of Z-modules.

(a) Prove that in MOD, f : M ! N is an epimorphism if and

only if f is a surjection.

(b) In ALG, does the equivalence of epimorphism and surjec-

tion hold? If yes, prove the equivalence, and if no, give a

counterexample (as simple as possible).

(6) Let G be a group with a normal subgroupN = hy, zi isomorphic

to (Z/2)2. Suppose that G has a subgroup Q = hxi isomorphic

to the cyclic group Z/3 such that the composition Q ⇢ G !G/N is an isomorphism. Finally, suppose that xyx�1

= z and

xzx�1= yz. Compute the character table of G.

(7) Let B be a commutative noetherian ring, and let A be a noe-

therian subring of B. Let I be the nilradical of B. If B/Iis finitely generated as an A-module, show that B is finitely

generated as an A-module.

(8) Let F be a field that contains the real numbers R as a sub-

field. Show that the tensor product F ⌦R C is either a field or

isomorphic to the product of two copies of F , F ⇥ F .

(9) Show that there is no simple group of order 616.

1

2

(10) By one definition, a Dedekind domain is a commutative noe-

therian integral domain R, integrally closed in its fraction field,

such that R is not a field and every nonzero prime ideal in

R is maximal. Let R be a Dedekind domain, and let S be a

multiplicatively closed subset of R. Show that the localization

R[S�1] is either the zero ring, a field, or a Dedekind domain.

ALGEBRA QUALIFYING EXAM

2018 SEPTEMBER 11

Instructions: Solve 10 of the 12 problems. If you solve more than 10, indicate

clearly which 10 you want graded. State clearly the theorems you use.

Write your University ID number (not your name) on every page you turn in.

* * *

Problem 1. Let Q8 = {±1,±i,±j,±k} be the quaternion group of order 8.

(1) Show that every non-trivial subgroup of Q8 contains �1.

(2) Show that Q8 does not embed in the symmetric group S7 (as a subgroup).

Problem 2. Let G be a finitely generated group having a subgroup of finite in-

dex n > 1. Show that G has finitely many subgroups of index n and has a proper

characteristic subgroup (i.e. preserved by all automorphisms) of finite index.

Problem 3. Let K/F be a finite extension of fields. Suppose that there exist

finitely many intermediate fields K/E/F . Show that K = F (x) for some x 2 K.

Problem 4. Let K be a subfield of the real numbers and f an irreducible degree 4

polynomial over K. Suppose that f has exactly two real roots. Show that the

Galois group of f is either S4 or of order 8.

Problem 5. Let R be a commutative ring. Show the following:

(a) Let S be a non-empty saturated multiplicative set in R, i.e. if a, b 2 R, then

ab 2 S if and only if a, b 2 S. Show that Rr S is a union of prime ideals.

(b) If R is a domain, show that R is a UFD if and only if every nonzero prime ideal

in R contains a non-zero principal prime ideal.

Problem 6. Let A be an integrally closed Noetherian domain with quotient field

F and K/F be a finite separable field extension.

(a) If {x1, . . . , xn} is a basis for K as an F -vector space, show that there exists

{y1, . . . , yn} in K such that TrK/F (xiyj) = �i,j for all i, j.(b) If B is the integral closure of A in K, show that B is a finitely generated

A-module.

Problem 7. Let F : C ! D be a functor with a right adjointG. Show that F is fully

faithful if and only if the unit of the adjunction ⌘ : IdC ! GF is an isomorphism.

Problem 8. Give an example of a diagram of commutative rings whose colimit in

the category of commutative rings is di↵erent from its colimit in the larger category

of rings (and ring homomorphisms).

Problem 9. Let f : M ! N and g : N ! M be two R-linear homomorphisms of

R-modules such that idM �gf is invertible. Show that idN �fg is invertible as well

and give a formula for its inverse. [Hint: You may use Analysis to make a guess.]

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2 ALGEBRA QUAL 2018 FALL

Problem 10. Consider the real algebra A = R[x, y] = R[X,Y ]/X2+Y 2� 1 where

x and y are the classes of X and Y respectively. Let M = A(1 + x) + Ay be the

ideal generated by 1 + x and y. (This is the Mobius band.)

(1) Show that there is an A-linear isomorphism A2 ⇠! M � M mapping the

canonical basis to (1 + x, y) and (�y, 1 + x).(2) Show that there is an A-linear isomorphism A

⇠! M ⌦A M mapping 1

to ((1 + x)⌦ (1 + x)) + (y ⌦ y).

Problem 11. Let G be a finite group, ! be a primitive 3rd root of 1 in C and

suppose that the complex character table of G contains the row

1 ! !21.

Determine the whole complex character table of G, the order of the group and the

order of its conjugacy classes.

Problem 12. Let F be a finite field and K ⇢ F the subfield of an algebraic closure

generated by all roots of unity. Find all simple finite dimensional K-algebras.

ALGEBRA QUALIFYING EXAM

Instructions: Do the following ten problems. Write your UID numberonly, not your name.

1. Let G be a finite solvable group and 1 = N ⊂ G be a minimal normalsubgroup. Prove that there exists a prime p such that N is either cyclic oforder p or a direct product of cyclic groups of order p.

2. An additive group (abelian group written additively) Q is called divisibleif any equation nx = y with 0 = n ∈ Z, y ∈ Q has a solution x ∈ Q. Let Q bea divisible group and A is a subgroup of an abelian group B. Give a completeproof of the following: every group homomorphism A → Q can be extendedto a group homomorphism B → Q.

3. Let d > 2 be a square-free integer. Show that the integer 2 in Z[√−d] is

irreducible but the ideal (2) in Z[√−d] is not a prime ideal.

4. Let R be a commutative local ring and P a finitely generated projectiveR-module. Prove that P is R-free.

5. Let Φn denote the nth cyclotomic polynomial in Z[X] and let a be apositive integer and p a (positive) prime not dividing n. Prove that if p | Φn(a)in Z, then p ≡ 1 mod n.

6. Let F be a field of characteristic p > 0 and a ∈ F×. Prove that if thepolynomial f = Xp − a has no root in F , then f is irreducible over F .

7. Let F be a field and let R be the ring of 3× 3 matrices over F with (3, 1)and (3, 2) entry equal to 0. Thus,

R =

⎡

⎣F F FF F F0 0 F

⎤

⎦ .

(a) Determine the Jacobson radical J of R.

(b) Is J a minimal left (respectively, right) ideal?

8. Prove that every finite group of order n is isomorphic to a subgroup ofGLn−1(C).

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9. a) Find a domain R and two nonzero elements a, b ∈ R such that R isequal to the intersection of the localizations R[1/a] and R[1/b] (in the quotientfield of R) and aR + bR = R.

b) Let C be the category of commutative rings. Prove that the functor C →Sets taking a commutative ring R to the set of all pairs (a, b) ∈ R2 such thataR + bR = R is not representable.

10. Let C be an abelian category. Prove that TFAE:

(1) Every object of C is projective.

(2) Every object of C is injective.

Algebra Qualifying ExamFall 2019

Write your University ID number (not your name) on every page.

1. Show that every group of order 315 is the direct product of a group of order 5

with a semidirect product of a normal subgroup of order 7 and a subgroup of order

9. How many such isomorphism classes are there?

2. Let L be a finite Galois extension of a field K inside an algebraic closure K of K.

Let M be a finite extension of K in K. Show that the following are equivalent:

i. L \M = K,

ii. [LM : K] = [L : K][M : K],

iii. every K-linearly independent subset of L is M -linearly independent.

3. Let I be the ideal (x2 � y2 + z2, (xy + 1)2 � z, z3) of R = C[x, y, z]. Find the

maximal ideals of R/I, as well as all of the points on the variety

V (I) = {(a, b, c) 2 C3 | f(a, b, c) = 0 for all f 2 I}.

4. Find all isomorphism classes of simple (i.e., irreducible) left modules over the ring

Mn(Z) of n-by-n matrices with Z-entries with n � 1.

5. Let R be a nonzero commutative ring. Consider the functor tB from the category

of R-modules to itself given by taking the (right) tensor product with an R-module

B.

a. Prove that tB commutes with colimits.

b. Construct an R-module B (for each R) such that tB does not commute with

limits in the category of R-modules.

6. Classify all finite subgroups of GL(2,R) up to conjugacy.

7. Let G be the group of order 12 with presentation

G = hg, h | g4 = 1, h3 = 1, ghg�1= h2i.

Find the conjugacy classes of G and the values of the characters of the irreducible

complex representations of G of dimension greater than 1 on representatives of these

classes.

8. Let M be a finitely generated module over an integral domain R. Show that there

is a nonzero element u 2 R such that the localization M [1/u] is a free module over

R[1/u].

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9. Let A be a unique factorization domain which is a Q-algebra. LetK be the fraction

field of A. Let L be a quadratic extension field of K. Show that the integral closure

of A in L is a finitely generated free A-module.

10. Compute the Galois groups of the Galois closures of the following field exten-

sions:

a. C(x)/C(x4 + 1),

b. C(x)/C(x4 + x2 + 1),

where C(y) denotes the field of rational functions over C in a variable y.

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