Do six problems

- 1.
- Let
*R*be a principal ideal domain. Assume that*M*is a nonzero finitely generated*R*-module with the property that the intersection of any two nonzero submodules is nonzero. Prove that*M**R*/*Rt*where*t*is either zero or some power of an irreducible element in*R*. - 2.
- Let
*G*be a group which acts transitively on a finite set*X*. Assume that there is an element*x*_{0}*X*whose stabilizer has no element of finite order other than 1.- (a)
- Show that if
*f**G*has finite order larger than 1, then*f*has no fixed points. - (b)
- Show that if the order of
*f**G*is a prime*q*, then |*X*| 0 mod*q*.

- 3.
- Let
*S*be a commutative ring with prime ideals*P*_{1},*P*_{2},...,*P*_{t}. Show that if*S*/*P*_{1}*P*_{2}...*P*_{t}is a finite set, then each of the*P*_{i}is a maximal ideal. - 4.
- Let
*p*be a prime. Prove that if every nontrivial finite field extension of the field*F*has degree divisible by*p*, then every finite field extension of*F*has degree a power of*p*. (You may assume that char*F*= 0.) - 5.
- Let
*R*be a commutative ring with a 1. If*M*and*N*are*R*-modules, then Hom(*M*,*N*) denotes the set of all*R*-module homomorphisms from*M*to*N*. If*f*:*N*- >*N'*is an*R*-module homomorphism, then*f*_{*}: Hom(*M*,*N*) - > Hom(*M*,*N'*) is defined by*f*_{*}(*g*) =*f*`o`*g*, the composition of*g*followed by*f*. Prove that*M*is a projective*R*-module if and only if for all surjective*f*:*N*- >*N'*, the function*f*_{*}is surjective. - 6.
- Let
*D*be a finite dihedral group, and let*V*be a finite dimensional complex vector space which is a*D*-module. (You may regard*D*as a group of linear transformations from*V*to itself.) Prove that if the only*D*-invariant subspaces of*V*are 0 and*V*itself (i.e.*V*is a simple or irreducible*D*-module), then dim_{}*V*2. - 7.
- Determine the Galois group of (the splitting field for) the
polynomial
*X*^{10}- 1 over the rational numbers.