Department of Mathematics and Statistics University at Albany

Preliminary Ph.D. Examination in Algebra

June 16, 1995

[10] 1.

Let Q denote the field of rational numbers. Prove or disprove one of the following assertions concerning an arbitrary symmetric matrix M with entries in Q. There exists an invertible matrix P with entries in Q such that:

(a) P M ^{t}P is diagonal.
(b) P MP^{−1} is diagonal.
[10] 2.  Let k be a field.
(a) Show that (x + 1) is a maximal ideal in the polynomial ring k[x].
(b) Show that (x + 1, y − 2) is a maximal ideal of k[x, y].

(c)

Let A be the quotient ring k[x, y]/(x + 1), and let φ be the k[x, y]-linear endomorphism of A given by

φ(f) = (y − 2)f .

Show that the cokernel of φ is a 1-dimensional k-module.

[10] 3.

Show that every automorphism of the symmetric group S_{3} (the group of all permutations of a set with 3 members) is an inner automorphism.

[10] 4.

Let E be a (finite) Galois extension field of F with Galois group G; let K be an intermediate field and H the subgroup of G that fixes each member of K. Show that the subgroup of G consisting of all σ in G for which σ(K) = K is the normalizer of H in G, i.e., the largest subgroup N of G containing H for which H is a normal subgroup of N.

[10] 5.

For K any field GL(n, K) denotes the group of invertible n × n matrices in the field K, and SL(n, K) denotes the group of such matrices of determinant 1. Prove that GL(n, K) is isomorphic to a semi-direct product of SL(n, K) with GL(1, K).

[10] 6.

Let R be a commutative ring with a unique maximal ideal M. Show that if e^{2} = e, then e = 0 or e = 1.

[10] 7.

For I and J ideals in a commutative ring R, prove that the natural R-algebra homomorphism

R/I ⊗_{R} R/J ⟶ R/(I + J)

is an isomorphism of R-algebras.

[10] 8.

Let K be the splitting field over the field Q of rational numbers of the polynomial

t^{4} + 4t^{2} + 2 .

Find the Galois group of K over Q.