Algebra Prelim, December 2007
Do all problems
- Let p and q be distinct prime integers.
- List all nonisomorphic abelian groups of order p3q, listing only
one for each isomorphism class.
- Show that if G is an abelian group of order p3q such that G
cannot be generated by one element but G can be generated by two
- Let K be a finite field extension of k, let
let f (x) be the irreducible polynomial of f over k. Prove that
n | deg f (x), then
n | [K : k].
- Let R be a UFD with quotient field Q and let f (x) be an
irreducible polynomial of degree
≥1 in R[x]. Let I denote
the ideal in Q[x] generated by f (x). Prove that Q[x]/I is a
- Prove that S4 is solvable.
- Let R be a ring with a 1 and let P and Q be projective
R-modules. Prove that if
f : P -> Q is a surjective
R-module homomorphism, then ker f is a projective R-module.
- Let R be an integral domain with quotient field Q. Show that if
V is a finite dimensional vector space over Q, then
(Q⊗RQ)⊗QV≌V as vector spaces over Q.
- Let K be a Galois extension of a field F of order 114. Prove
that there are intermediate fields
F = K0⊆K1⊆K2⊆K3⊆K4 = K such that
[Ki : Ki-1] = 11
and Ki is a Galois extension of F, for
i = 1, 2, 3, 4.
- Let R be a local commutative ring with 1 and with maximal ideal
M. Suppose I is an ideal such that
Prove that R/I is not a projective R-module.