Do all problems

- Let
*G*be a group of order 105 with a normal Sylow 3-subgroup. Show that*G*is abelian. - Find the Galois group of
*f*(*x*) =*x*^{4}-2*x*^{2}- 2 over**Q**, describing its generators explicitly as permutations of the roots of*f*. - Let
*p*be a prime. Prove that the extension**F**_{pn}⊃**F**_{p}has Galois group generated by the Frobenius automorphism σ :**F**_{pn}->**F**_{pn}given by σ(*a*) =*a*^{p}for all*a*∈**F**_{pn}. - Show that there is no 3 by 3 matrix
*A*with entries in**Q**, such that*A*^{8}=*I*but*A*^{4}≠*I*. - Let
*R*be an integral domain. A nonzero nonunit element*p*∈*R*is*prime*if*p*|*ab*implies*p*|*a*or*p*|*b*. A nonzero nonunit element*p*∈*R*is*irreducible*if*p*=*ab*implies*a*or*b*is a unit. Show that- (a)
- Every prime is irreducible.
- (b)
- If
*R*is a UFD, then every irreducible is prime.

- Let
*R*be the ring**Z**/6**Z**and let*I*be the ideal 3**Z**/6**Z**. Prove that*I*⊗_{R}*I*≌*I*as*R*-modules. - Let
*S*be a multiplicatively closed nonempty subset of the commutative ring*R*with a 1. Assume that 0*S*.- (a)
- Show that if
*R*is a PID, then*S*^{-1}*R*is a PID. - (b)
- Show that if
*R*is a UFD, then*S*^{-1}*R*is a UFD.

- Let
*R*be a commutative ring with a 1.- (a)
- Show that if
*x*∈*R*is nilpotent and*y*∈*R*is a unit in*R*, then*x*+*y*is a unit in*R*. - (b)
- Let
*f*=*a*_{0}+*a*_{1}*x*+*a*_{2}*x*^{2}+ ... +*a*_{n}*x*^{n}∈*R*[*x*]. Show that*f*is a unit in*R*[*x*] if an only if*a*_{0}is a unit in*R*and*a*_{i}is nilpotent for*i*> 0.

Peter Linnell 2010-12-20