Math 901-902 Comprehensive Exam – June 2015
Instructions. Each problems is worth 20 points.
Do one problem from each one of the six subsections.
If you have doubts about the wording of a problem, please ask for clarification. In a
multi-part problems you can use the assertions of earlier parts.
I. Representation Theory. In this part k denotes a field and A a (unital) k-algebra.
Representations of Algebras. Do one of the following two problems.
1. Prove the following statements for a representation V of A.
(a) If dimk V is finite and V 6= 0, then V contains an irreducible representation.
(b) Does the conclusion of (a) hold when dimk V = ∞? Justify your answer.
In the next two parts V is assumed to be finite-dimensional and irreducible.
(c) If k is algebraically closed and α : V → V is a homomorphism of representations,
then there is an element a ∈ k, such that α(v) = av for all v ∈ V .
(d) Does the conclusion of (c) hold when k is not algebraically closed?
Justify your answer.
2. Let U and V be irreducible representations of A, and W = U ⊕ V .
Let U 0 and V 0 are subrepresentations of W , such that W = U 0 ⊕ V 0 .
Prove the following statements.
(a) If U ∼
6 V , then {U 0 , V 0 } = {U, V }; that is, either U 0 = U and V 0 = V , or U 0 = V
=
and V 0 = U .
(b) If U ∼
6 {U, V }.
= V , then there is a decomposition with {U 0 , V 0 } =
(c) If U ∼
= V and k is infinite, then there exist infinitely many decompositions with
U 0 , V 0 6∈ {U, V }.
Representations of Finite Groups. Do one of the following two problems.
3. Let λ : G → GL(V ) a finite-dimensional representation of a finite abelian group G.
(a) Prove that if char(k) = 0, then there exits a basis of V in which the matrix of
λ(g) is diagonal for every g ∈ G.
(b) Does the conclusion of (a) hold if char(k) > 0? Justify your answer.
(c) Does the conclusion of (a) hold if k is not algebraically closed? Justify your
answer.
4. Let a = (123), b = (456) and c = (23)(56) be permutations in the symmetric group S6 .
The group G = ha, b, ci then has order 18.
(a) Prove that the subgroups of G, listed below, are normal:
A = hai B = hbi C = habi D = hab−1 i
Assume that k is algebraically closed of characteristic zero.
Let {V1 , . . . , Vn } be the irreducible representations of G/H for H = A, . . . , D, with
dimk Vi > 1, viewed as representations of G through the homomorphisms G → G/H.
(b) Prove that these representations are pairwise non-isomorphic and find n.
(c) List all the irreducible representations of G.
Complex characters of Finite Groups. Do one of the following two problems.
5. Write down the character table of the quaternion group Q8 . Explain your work.
6. Let G be a finite group, χ a character of G, and g an element of order n in G.
Prove the following statements.
(a) χ(g) is a sum of complex roots of unity of order n.
(b) If n = 2, then χ(g) is an integer, and χ(g) ≡ χ(1) mod 2.
1
2
II. Commutative Algebra. In this part k denotes a field and R a commutative ring.
Gröbner Bases and Primary Decomposition. Do one of the following two problems.
7. Choose the lexicographic order with x > y on the polynomial ring k[x, y].
(a) Prove that {x3 − y, x2 y − y 2 , xy 2 − y 2 , y 3 − y 2 } is the reduced Gröbner basis of
the ideal I = (−x3 + y, x2 y − y 2 ).
(b) Determine whether the polynomial f = x6 − x5 y is in I.
8. Let R be a noetherian ring, M a maximal ideal, and I an M -primary ideal.
Prove the following statements.
(a) If I = Q1 ∩ · · · ∩ Qs is an irredundant decomposition as intersection of irreducible
ideals, then s is equal to the length of the R-module (I : M ) = {x ∈ M | xM ⊆ I}.
(b) An M -primary ideal I 6= M is irreducible if and only if there is an ideal J, such
that I ( J ⊆ M holds and J is contained in every ideal K satisfying I ( K ⊆ M .
Integral Extensions. Do one of the following two problems.
9. Let k be a field, k[x, y] a polynomial ring in x, y, and P the ideal (y 2 − x3 − x2 ).
(a) Prove that P is prime.
(b) Find the normalization of the ring R = k[x, y]/P .
10. Let S be an integral domain, G be a finite group of automorphisms of S, and R = S G
the fixed ring of G.
Prove the following statements.
(a) The ring S is an integral extension of R.
(b) If Q is a prime ideal of S, then so is g(Q) for every g ∈ G.
(c) For every P ∈ Spec(R) the group G acts transitively on the set
{Q ∈ Spec(S) | R ∩ Q = P }
Krull Dimension. Do one of the following two problems.
11. Let k be a field and R = k[[x]] the ring of formal power series in x.
Prove the following statements.
(a) The ring R is a principal ideal domain.
(b) If I and J are ideals of R, then one is contained in the other.
(c) Find in the ring S = R[y] maximal ideals of height 1 and of height 2.
12. Let R be an artinian local ring and M its maximal ideal.
Prove that the following conditions are equivalent.
(i) Every ideal of R is principal.
(ii) The maximal ideal M is principal.
(iii) Every proper ideal of R is equal to M i for some i ≥ 1.
(iv) Every two ideals I and J of R satisfy I ⊆ J or I ⊇ J.