Answer all questions

- Let
*R*be a commutative ring with unity and let*I*,*J*be ideals of*R*.- Prove that the product
*IJ*= {*x**R*|*x*= S_{i = 1}^{n}*a*_{i}*b*_{i}with*a*_{i}*I*and*b*_{i}*J*} is an ideal of*R*. - Prove that
*IJ**I**J*. - If
*I*+*J*=*R*, prove that*IJ*=*I**J*. - If
*IJ*=*I**J*for all ideals of*R*and*R*is an integral domain, prove that*R*is a field. (Hint: let*I*=*Ra*where*a*0 be a principal ideal or*R*.)

- Prove that the product
- Let
*F*be a finite Galois extension of the field*K*with Gal(*F*/*K*)*S*_{5}.- Show that there are more than 40 fields strictly between
*F*and*K*. - Show that there is a unique proper subfield
*E*of*F*with*E**K*such that*E*/*K*is a Galois extension. Determine [*E*:*K*] and describe Gal(*E*/*K*) up to isomorphism.

- Show that there are more than 40 fields strictly between
- Let
*G*be a group of order 455.- Prove that
*G*is not simple. - Prove that
*G*is cyclic.

- Prove that
- Let
*R*be a PID, let*n*be a positive integer, and let*A*and*B*be finitely generated*R*-modules. If*A*^{n}*B*^{n}, prove that*A**B*. (*A*^{n}denotes the direct sum of*n*copies of*A*.) - Let
*P*be a finitely generated projective -module. If*P*is also injective, prove that*P*= 0. - Let
*A*,*B*be abelian groups, and let*m*be a positive integer. Prove that*A*(*B*/*mB*) (*A**B*)/*m*(*A**B*). - Prove that a group of order 588 is solvable.
- Let
*K*= ( + ,).- Determine
[
*K*: ]. - Compute
Gal(
*K*/).

- Determine
[