Algebra Prelim, December 2007

Do all problems

1. Let p and q be distinct prime integers.

   (a)

   List all nonisomorphic abelian groups of order p³q, listing only one for each isomorphism class.

   (b)

   Show that if G is an abelian group of order p³q such that G cannot be generated by one element but G can be generated by two elements, then G≌Z_p²×Z_pq.

2. Let K be a finite field extension of k, let α∈K, and let f (x) be the irreducible polynomial of f over k. Prove that if n | deg f (x), then n | [K : k].

3. Let R be a UFD with quotient field Q and let f (x) be an irreducible polynomial of degree ≥1 in R[x]. Let I denote the ideal in Q[x] generated by f (x). Prove that Q[x]/I is a field.

4. Prove that S₄ is solvable.

5. Let R be a ring with a 1 and let P and Q be projective R-modules. Prove that if f : P -> Q is a surjective R-module homomorphism, then ker f is a projective R-module.

6. Let R be an integral domain with quotient field Q. Show that if V is a finite dimensional vector space over Q, then (Q⊗_RQ)⊗_QV≌V as vector spaces over Q.

7. Let K be a Galois extension of a field F of order 11⁴. Prove that there are intermediate fields F = K₀⊆K₁⊆K₂⊆K₃⊆K₄ = K such that [K_i : K_i-1] = 11 and K_i is a Galois extension of F, for i = 1, 2, 3, 4.

8. Let R be a local commutative ring with 1 and with maximal ideal M. Suppose I is an ideal such that 0⊊I⊆M. Prove that R/I is not a projective R-module.