Algebra Prelim, January 2005
Do all problems
- If p and q are distinct primes and G is a finite group of order
p2q, prove that G has a nontrivial normal Sylow subgroup.
- Find the Galois group of K over the rationals
K is the splitting field of the polynomial
x4 +4x2 + 2.
- Show that
Q is not a projective
- Let R be a commutative Noetherian ring with a 1 and let M be a
finitely generated R-module. Show that if
f : M - > M is a
surjective R-module homomorphism, then it must also be injective.
(Hint: consider the kernels of fn.)
- Suppose R is a principal ideal domain that is not a field, and that
M is a finitely generated R-module. Suppose further that for
every irreducible element
p e R, the R/pR-module M/pM is
cyclic (has a single generator). Show that M is cyclic.
- Let G be a finite group with a composition series of length 2.
Prove that if M and N are distinct nonidentity proper normal
subgroups of G, then
G = M X N.
- Let R be the ring
Q + x2Q[x], the collection
of all polynomials with rational coefficients that have no x term.
- Show that if
0 =/= f e R, then R/fR is a finite dimensional
vector space over
- Use part (a) to prove that every nonzero prime ideal of R is