Algebra Prelim, August 2009

Do all problems

1. Let p be a prime, let n be a non-negative integer, and let S be a set of order pn. Suppose G is a finite group that acts transitively by permutations on S (so if s, tS, then there exists gG such that gs = t) and let P be a Sylow p-subgroup of G. Prove that P acts transitively on S.

2. Prove that there is no simple group of order 448 = 7*64.

3. (a)
Prove that x2 + 1 is irreducible in Z/3Z[x].

(b)
Prove that x3 +3x2 - 9x + 12 is irreducible in Z[i][x].

4. Let R be a PID, let M be a finitely generated right R-module, and let N be an R-submodule of M. Prove that there exists an R-submodule L of M and 0≠rR such that LN = 0 and MrL + N.

5. Let R be a ring and suppose we are given a commutative diagram of R-modules and homomorphisms,


f
A -------------> B
|                |
|                |
g|                |h
|                |
|                |
\/      j        \/
C -------------> D



where f is onto and j is one-to-one. Prove that there exists a unique R-module homomorphism k : B -> C such that the resulting diagram commutes (so kf = g and jk = h).

6. Compute the isomorphism type of the Galois group of x4 -2x2 + 9 over Q.

7. Let R be a commutative ring with a 1 and let I, J be ideals of R. Prove that R/IRR/JR/(I + J) as R-modules.

Peter Linnell 2009-08-07