Do all problems

- Let
*p*be a prime, let*n*be a non-negative integer, and let*S*be a set of order*p*^{n}. 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*. - Prove that there is no simple group of order
448 = 7*64.
- (a)
- Prove that
*x*^{2}+ 1 is irreducible in**Z**/3**Z**[*x*]. - (b)
- Prove that
*x*^{3}+3*x*^{2}- 9*x*+ 12 is irreducible in**Z**[*i*][*x*].

- 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*. - 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*). - Compute the isomorphism type of the Galois group of
*x*^{4}-2*x*^{2}+ 9 over**Q**. - Let
*R*be a commutative ring with a 1 and let*I*,*J*be ideals of*R*. Prove that*R*/*I*⊗_{R}*R*/*J*≌*R*/(*I*+*J*) as*R*-modules.

Peter Linnell 2009-08-07