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

1. Let G be a group, let [G, G] be the commutator subgroup of G, and let N be the subgroup of G generated by the set

K = {x^{2} : x ∈ G}.

(a) Prove that N is a normal subgroup of G, and that [G, G] ⊂ N.

(b) Give an example of a group G with [G, G] ≠ N.

(c) Give an example of a group G with K ≠ N.

2. Show that ℤ[x] is a unique factorization domain. Is it a principal ideal domain? Prove your answer.

3. Let R be a ring.

(a) Define what it means for a R-module P to be projective in terms of exact

sequences of R-modules

M′ → M → 0

and module homomorphisms f : P → M.

(b) Using the definition from (a), prove that an R-module P is projective if and only if P is a direct summand of a free R-module.

4. Let K/F be an extension of fields and f an irreducible polynomial over F. Prove that if u and v are roots of f in K then there is an F -homomorphism σ : F (u) → F (v). Show by example that the result may be false if f is not irreducible.

5. Let f(x) = x^{4} − 4.

(a) Determine the degree of the splitting field K of f(x) over ℚ.

(b) Determine all the subextensions F such that ℚ ⊂ F ⊂ K, and arrange these in a graph ordered by inclusion.

6. Determine all possible operations of the symmetric group S_{3} on a set of 4 elements. (Two such operations will be considered the same if by relabelling the 4 elements the operations become identical.)