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Doctoral Algebra Examination

Oklahoma State University, Department of Mathematics August, 1995

Give clear and coherent proofs of all your assertions. If you cannot solve a problem completely, state the definitions of all terms in the problem and record any partial progress you may have made.

1. Use the Sylow theorems to prove that any group of order 15 is cyclic.

2. Let G, H, and K be groups.

(a) Define what it means for a sequence of group homomorphisms

H ⟶^{α} G ⟶^{β} K

to be exact.

(b) Define the term short exact sequence.

(c) Give an example of a short exact sequence

1 ⟶ H ⟶^{α} G ⟶^{β} K ⟶ 1

where G is not isomorphic to the direct product H × K.

3. Let R be a ring.

(a) Define what it means for an R-module to be injective.

(b) Prove that any homomorphic image of an injective ℤ-module is injective.

4. Show that the intersection of all prime ideals in a commutative Noetherian ring is

nilpotent (i.e. 𝔞 is nilpotent if 𝔞^{k} = 0 for some integer k ≥ 1). (Next page)

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5. Let K_{1}, K_{2} be finite extensions of the field k, all of which are contained in the field L. (a) Define the compositum K_{1}K_{2}.

(b) Prove that [K_{1}K_{2} : k] ≤ [K_{1} : k] · [K_{2} : k].

(c) Prove that, if the degrees [K_{1} : k] and [K_{2} : k] are relatively prime, then equality

holds in part (b).

6. Let K be a field with p^{n} elements where p is a prime and n is a positive integer. Show that every element of K has a unique p-th root in K.

7. Let F be a field and let L = F (x_{1}, …, x_{n}) be the field of all rational functions in the indeterminates x_{1}, …, x_{n} with coefficients in F. Let the symmetric group S_{n} act on f(x) ∈ L by (σ · f)(x_{1}, …, x_{n}) = f(x_{σ(1)}, …, x_{σ(n)}).

(a) Show that this identifies S_{n} as a group of field automorphisms of L.

(b) Let K be the subfield of L fixed by the action of all elements of S_{n} (K is the field of symmetric functions). Show that L is the splitting field of an irreducible polynomial of degree n over K (i.e. with coefficients in K).

(c) Prove that L/K is Galois with group S_{n}.