Do all problems

- Recall that a proper subgroup of the group
*G*is a subgroup*H*of*G*with*G*≠*H*. Now suppose*G*is a finite cyclic group. Prove that*G*is not a union of proper subgroups. - Prove that a group of order
6435 = 9·5·11·13
cannot be simple.
- Prove that
M
_{2}(**Q**)⊗_{M2(Z)}M_{2}(**Q**)≌M_{2}(**Q**) as (M_{2}(**Q**), M_{2}(**Q**))-bimodules ( M_{2}(**Q**) indicates the ring of 2 by 2 matrices with entries in**Q**). - Let
*R*be a PID which is not a field and let*M*be a finitely generated injective*R*-module. Prove that*M*= 0. - Let
*p*be an odd prime and for a positive integer*n*, let ζ_{n}=*e*^{2πi/n}, a primitive*n*th root of 1.- (a)
- Prove that
**Q**(ζ_{p}) =**Q**(ζ_{2p}). - (b)
- Prove that
1 +
*x*^{2}+*x*^{4}+ ... +*x*^{2p-2}is the product of two irreducible polynomials in**Q**[*x*].

- Determine the isomorphism class of the Galois group of the polynomial
*x*^{5}- 5*x*- 1 over**Q**. - For
*n*a positive integer, let**A**^{n}denote affine*n*-space over**Q**.- (a)
- Prove that every element of
**Q**[*x*,*y*]/(*x*^{3}-*y*^{2}) can be written in the form (*x*^{3}-*y*^{2}) +*f*(*x*) +*yg*(*x*) where*f*(*x*),*g*(*x*)∈**Q**[*x*]. - (b)
- Prove that
**Q**[*x*,*y*]/(*x*^{3}-*y*^{2})≌**Q**[*t*^{2},*t*^{3}], the subring of the polynomial ring**Q**[*t*] generated by*t*^{2},*t*^{3}. - (c)
- Prove that
**Q**[*t*^{2},*t*^{3}] is not a UFD. - (d)
- Let
*V*denote the affine algebraic set Z(*x*^{3}-*y*^{2}), the zero set of*x*^{3}-*y*^{2}in**A**^{2}. Determine the coordinate ring of*V*. - (e)
- Is
*V*isomorphic to**A**^{1}as affine algebraic sets? Justify your answer.

Peter Linnell 2012-01-11