Department of Mathematics and Statistics
University at Albany
Preliminary Ph.D. Examination in Algebra
June 16, 1995
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:
PM tP is diagonal.
PMP−1 is diagonal.
Let k be a field.
Show that (x + 1) is a maximal ideal in the polynomial ring k[x].
Show that (x + 1, y − 2) is a maximal ideal of k[x, y].
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.
Show that every automorphism of the symmetric group S3 (the group of all
permutations of a set with 3 members) is an inner automorphism.
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 .
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
Let R be a commutative ring with a unique maximal ideal M . Show that if
e2 = e, then e = 0 or e = 1.
For I and J ideals in a commutative ring R, prove that the natural R-algebra
R/I ⊗R R/J −→ R/(I + J)
is an isomorphism of R-algebras.
Let K be the splitting field over the field Q of rational numbers of the polynomial
t4 + 4t2 + 2 .
Find the Galois group of K over Q.