Do all problems

- How many elements of order 7 must there be in a simple group of order
168?
- Let
ρ be a primitive 4th root of 1 over
**Q**.- (a)
- Compute the Galois group of
(
*x*^{4}-2)(*x*^{2}- 3) over**Q**and**Q**(ρ). - (b)
- Is
**Q**(ρ) Galois over**Q**? (Explain your answer.) - (c)
- Are there any proper subfields of the splitting field of
(
*x*^{4}-2)(*x*^{2}- 3) over**Q**(ρ) that are Galois over**Q**(ρ)? (Explain your answer.)

- Suppose that
0 -->
*A*--^{f}-->*B*--^{g}-->*C*--> 0 is a split exact sequence of left*R*-modules, where*R*is a ring with a 1. If*D*is a right*R*-module, prove that 1*f*:*D**A*-->*D**B*is a monomorphism. - Let
*R*be a PID and let*S*be a multiplicatively closed subset of*R*. Assume that*S*is nonempty and that*S*does not contain 0. Prove that*S*^{-1}*R*is a PID. - Prove that a group of order
2
^{4}·11^{2}is solvable. - Prove that
*f*(*x*) =*x*^{4}+ 9*x*- 30 is an irreducible polynomial over**Q**. - Let
*g*(*x*) =*x*^{2}+ 2 and let*I*be the ideal in**Q**[*x*] generated by the product*f*(*x*)*g*(*x*). Show that**Q**[*x*]/*I*is the product of two fields. What is the dimension over**Q**of these fields?

- Prove that
- Let
*R*be an integral domain. If*X*is an*R*-module, then let*t*(*X*) denote the subset {*x*∈*X*|*rx*= 0 for some nonzero*r*∈*R*}.- Prove that
*t*(*X*) is a submodule of*X*. - Prove that
*t*(*X*/*t*(*X*)) = 0. - Prove that if
*X*/*t*(*X*) is a nonzero cyclic*R*-module, then*X*is isomorphic to*t*(*X*)⊕*R*.

- Prove that

Peter Linnell 2007-08-11