Do all problems

- Let
*p*and*q*be distinct prime integers.- (a)
- List all nonisomorphic abelian groups of order
*p*^{3}*q*, listing only one for each isomorphism class. - (b)
- Show that if
*G*is an abelian group of order*p*^{3}*q*such that*G*cannot be generated by one element but*G*can be generated by two elements, then*G*≌**Z**_{p2}×**Z**_{pq}.

- 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*]. - 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. - Prove that
*S*_{4}is solvable. - 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. - Let
*R*be an integral domain with quotient field*Q*. Show that if*V*is a finite dimensional vector space over*Q*, then (*Q*⊗_{R}*Q*)⊗_{Q}*V*≌*V*as vector spaces over*Q*. - Let
*K*be a Galois extension of a field*F*of order 11^{4}. Prove that there are intermediate fields*F*=*K*_{0}⊆*K*_{1}⊆*K*_{2}⊆*K*_{3}⊆*K*_{4}=*K*such that [*K*_{i}:*K*_{i-1}] = 11 and*K*_{i}is a Galois extension of*F*, for*i*= 1, 2, 3, 4. - 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.

Peter Linnell 2009-08-06