Algebra Prelim, Fall 2005

Do all problems

1. Show that there are exactly 5 nonisomorphic groups of order 18.

2. Let A be a commutative ring and set B = A[X, Y]/(X² - Y²). Prove that A is a Noetherian ring if and only if B is a Noetherian ring.

3. Let F be a field with more than 2 elements and let GL₂(F) denote the group of 2 X 2 invertible matrices with entries in F. Consider the action of GL₂(F) on one-dimensional subspaces of F². Show that the stabilizer of a one-dimensional subspace is never simple.

4. Let R be the ring Z[X] and set M = 2R + XR. Prove or disprove: M is a free R-module.

5. 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.

6. Let S be a simple algebra of finite dimension n over C. Prove that there are $\sqrt{{n}}$ maximal left ideals of S whose intersection is zero.

7. 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 $\overline{{F}}$ be the algebraic closure of F. Show that if $\overline{{F}}$ $\otimes_{F}^{}$ L is a field, then F = L.