Ph.D. Algebra Examination Oklahoma State University Department of Mathematics June, 1998

1. Let p, q and r be prime numbers with r > pq. Prove that any group G of order pqr is solvable.

2. Let M be a submodule of the finitely generated module N over the principal ideal domain R. Prove that M is finitely generated.

3. Let R be a ring. Show that an R-module M is isomorphic to A_{1} ⨁ A_{2} if and only if there are R-homomorphisms f_{i} : A_{i} → M, i = 1, 2, such that for any module B and Rhomomorphisms g_{i} : A_{i} → B, i = 1, 2, there is a unique R-homomorphism h : M → B so that g_{i} = h ∘ f_{i}.

4. (a) For each prime p, show there exists a field 𝔽_{p} with p elements.

(b) For each n ≥ 2 show that there exists a field 𝔽_{q} with q = p^{n} elements.

(c) If p_{1} and p_{2} are distinct primes, does there exist a field with p_{1}p_{2} elements? Prove your answer.

5. Let L = ℚ(ζ_{13}) with ζ_{13} a primitive 13th root of unity.

(a) Prove that there exists a unique subfield K ⊂ L with [K : ℚ] = 4.

(b) Prove that K is Galois over ℚ and that the Galois group of K over ℚ is cyclic.

(c) Let α ∈ K be an element with K = ℚ(α). Prove that ℚ(α^{3}) ≠ ℚ.

(d) Prove that ℚ(α^{3}) = K.

6. Let G be a finite group. G is metabelian if there exists an abelian normal subgroup H of G such that G/H is abelian.

(a) Prove that if G is metabelian and K is a homomorphic image of G then K is metabelian.

(b) Prove that if A is a subgroup of a metabelian group then A is metabelian.