Algebra Prelim, January 1999

Answer all questions

1.

Let R be a commutative ring with a 1$\ne$ 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 $\in$ R \ 0 such that rs = 0. A unit in R is an element r such that there exists s $\in$ R such that rs = 1.)

4.

Let K be a subfield of the field L and let a $\in$ 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 $\in$ G | x^p = 1} has order p². Describe all possible groups G (up to isomorphism). Justify your answer.

6.

Let k $\subseteq$ K $\subseteq$ L be fields such that K is a splitting field over k, and let s $\in$ 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 2^n! fields between F and E.