Answer all questions

- 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*p*^{3}*q*^{3}. If*G*has a normal*p*-Sylow subgroup, prove that*G*has a normal subgroup*H*of order*p*^{3}*q*. - 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*(a^{2}) =*K*(a). - 5.
- Let
*p*be a prime and let*G*be an abelian group of order*p*^{6}. Suppose the set {*x**G*|*x*^{p}= 1} has order*p*^{2}. 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 2^{n!}fields between*F*and*E*.