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, t∈S, then there exists g∈G 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≠r∈R such that L∩N = 0 and Mr⊆L + 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/I⊗_RR/J≌R/(I + J) as R-modules.