Algebra Prelim, Fall 2004

Do all problems

1. 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 C2p of order 2p where p is an odd prime.

2. Let G be a simple group of order 4032 = 8!/10. Prove that G is not isomorphic to a subgroup of the alternating group A8. Deduce that G has at least 216 elements of order 7.

3. 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 bn e I for some positive integer n. Prove that if R is a PID, then I is primary if and only if I = Pn for some prime ideal P of R and some positive integer n.

4. Let k be a field and let k[x2, x3] denote the subring of the polynomial ring k[x] generated by k and {x2, x3}. 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.

5. 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.

6. 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.

7. Let K be a splitting field over the rational numbers Q of the polynomial x4 + 16. Determine the Galois group of K/Q.

Peter Linnell 2004-08-25