Do all problems

- An automorphism of a group is an isomorphism of the group with
itself. The set of automorphisms
Aut(
*G*) of a group*G*is itself a group under composition of functions. Find the group of automorphisms of the cyclic group*C*_{2p}of order 2*p*where*p*is an odd prime. - Let
*G*be a simple group of order 4032 = 8!/10. Prove that*G*is not isomorphic to a subgroup of the alternating group*A*_{8}. Deduce that*G*has at least 216 elements of order 7. - An ideal
*I*in a commutative ring*R*with unit is called*primary*if*I*=/=*R*and whenever*ab*e*I*and*a**I*, then*b*^{n}e*I*for some positive integer*n*. Prove that if*R*is a PID, then*I*is primary if and only if*I*=*P*^{n}for some prime ideal*P*of*R*and some positive integer*n*. - Let
*k*be a field and let*k*[*x*^{2},*x*^{3}] denote the subring of the polynomial ring*k*[*x*] generated by*k*and {*x*^{2},*x*^{3}}. Prove that every ideal of*R*can be generated by two elements. Hint: if the ideal is nonzero, we may choose one of the generators to be a polynomial of least degree. - Let
*k*be a field, let*f*e*k*[*x*] be a polynomial of positive degree and let*M*be a finitely generated*k*[*x*]-module. Suppose every element of*M*can be written in the form*fm*where*m*e*M*. Prove that*M*has finite dimension as a vector space over*k*. - Let
*R*be an integral domain (commutative ring with 1 =/= 0 and without nontrivial zero divisors) and suppose*R*when viewed as a left*R*-module is injective. Prove that*R*is a field. - Let
*K*be a splitting field over the rational numbers**Q**of the polynomial*x*^{4}+ 16. Determine the Galois group of*K*/**Q**.

Peter Linnell 2004-08-25