Instructions: do all problems.

- Let
*K*denote the splitting field over the rational numbers**Q**of the polynomial*f*(*x*) =*x*^{5}+*x*^{4}+ 3*x*+ 3.- (a)
- What is
[
*K*:**Q**]? - (b)
- Determine the Galois group
Gal(
*K*/**Q**).

- Prove or disprove: If
*x*and*y*are elements of a finite abelian group*G*with the same order, then there is an automorphism q of*G*such that q(*x*) =*y*. - Let
*H*be a group of order 2002. Prove that the number of elements in the set {*h**H*|*h*^{2}=*e*} is even (where*e*is the identity of*H*). - Let
*G*be a group with the following property: for each*g**G*- 1, there exists a normal subgroup*K*of*G*such that*G*/*K*is abelian and*g**K*(where 1 is the trivial subgroup). Prove that*G*is abelian. - Prove that if
*A*and*B*are commutative Noetherian rings, then so is the cartesian product*A*X*B*. - Let
*R*be a commutative ring, let*P*be a projective*R*-module and let*I*be an ideal in*R*. Prove*P*/*IP*is a projective*R*/*I*-module. - Let
*k*be a field and let*I*be an ideal of*k*[*x*] (where*k*[*x*] is the polynomial ring over*k*in the variable*x*).- (a)
- Show that
*k*+*I*is a subring of*k*[*x*]. - (b)
- Prove that if
*I*0 then*k*[*x*]*k*[*x*]/*I*is finite dimensional over*k*.