Algebra Prelim, January 1999

1.
Let R be a commutative ring with a 1 0. If every ideal of R except the ideal R is a prime ideal, prove that R is a field.
2.
Let p and q be distinct primes and let G be a group of order p3q3. If G has a normal p-Sylow subgroup, prove that G has a normal subgroup H of order p3q.

3.
Let R be a commutative ring with a 1. Prove that R is isomorphic to a proper R-submodule of R if and only if there exists an element in R which is neither a zero divisor nor a unit. (A zero divisor is an element r such that there exists s R \ 0 such that rs = 0. A unit in R is an element r such that there exists s R such that rs = 1.)

4.
Let K be a subfield of the field L and let a L. If [K(a) : K] is odd, prove that K(a2) = K(a).

5.
Let p be a prime and let G be an abelian group of order p6. Suppose the set {x G | xp = 1} has order p2. Describe all possible groups G (up to isomorphism). Justify your answer.

6.
Let k K L be fields such that K is a splitting field over k, and let s Gal(L/k). Prove that s(K) = K.

7.
Prove that there is no simple group of order 280.

8.
Let n be a positive integer, let E be a field of characteristic zero, and let F be a subfield of E such that [E : F] = n. Prove that there are at most 2n! fields between F and E.

Peter Linnell
1999-01-15