Do all problems

- Show that there are exactly 5 nonisomorphic groups of order 18.
- Let
*A*be a commutative ring and set*B*=*A*[*X*,*Y*]/(*X*^{2}-*Y*^{2}). Prove that*A*is a Noetherian ring if and only if*B*is a Noetherian ring. - Let
*F*be a field with more than 2 elements and let GL_{2}(*F*) denote the group of 2 X 2 invertible matrices with entries in*F*. Consider the action of GL_{2}(*F*) on one-dimensional subspaces of*F*^{2}. Show that the stabilizer of a one-dimensional subspace is never simple. - Let
*R*be the ring**Z**[*X*] and set*M*= 2*R*+*XR*. Prove or disprove:*M*is a free*R*-module. - Let
*F*be a field of characteristic zero. Suppose that*K*/*F*is finite Galois extension with Galois group*G*. Prove that if*a*e*K*and s(*a*) -*a*e*F*for all s e*G*, then*a*e*F*. - Let
*S*be a simple algebra of finite dimension*n*over**C**. Prove that there are maximal left ideals of*S*whose intersection is zero. - Recall that if
*F*is a field, then the tensor product of two*F*-algebras (over*F*) is another*F*-algebra. Let*L*be a finite field extension of*F*and let be the algebraic closure of*F*. Show that if*L*is a field, then*F*=*L*.

Peter Linnell 2005-08-20