Algebra Preliminary Exam, Spring 1993

Do six problems

1.

Let R be a principal ideal domain. Assume that M is a nonzero finitely generated R-module with the property that the intersection of any two nonzero submodules is nonzero. Prove that M $\cong$ R/Rt where t is either zero or some power of an irreducible element in R.

2.

Let G be a group which acts transitively on a finite set X. Assume that there is an element x₀ $\in$ X whose stabilizer has no element of finite order other than 1.

(a)

Show that if f $\in$ G has finite order larger than 1, then f has no fixed points.

(b)

Show that if the order of f $\in$ G is a prime q, then | X| $\equiv$ 0 mod q.

3.

Let S be a commutative ring with prime ideals P₁, P₂,..., P_t. Show that if S/P₁ $\cap$ P₂ $\cap$ ... $\cap$ P_t is a finite set, then each of the P_i is a maximal ideal.

4.

Let p be a prime. Prove that if every nontrivial finite field extension of the field F has degree divisible by p, then every finite field extension of F has degree a power of p. (You may assume that charF = 0.)

5.

Let R be a commutative ring with a 1. If M and N are R-modules, then Hom(M, N) denotes the set of all R-module homomorphisms from M to N. If f : N - > N' is an R-module homomorphism, then f_* : Hom(M, N) - > Hom(M, N') is defined by f_*(g) = fog, the composition of g followed by f. Prove that M is a projective R-module if and only if for all surjective f : N - > N', the function f_* is surjective.

6.

Let D be a finite dihedral group, and let V be a finite dimensional complex vector space which is a D-module. (You may regard D as a group of linear transformations from V to itself.) Prove that if the only D-invariant subspaces of V are 0 and V itself (i.e. V is a simple or irreducible D-module), then dim _{$\mathbb {C}$}V$\le$2.

7.

Determine the Galois group of (the splitting field for) the polynomial X¹⁰ - 1 over the rational numbers.