Do all problems

- 1.
- Find the Galois group of
*x*^{6}- 1 over . - 2.
- Show that a semisimple right Artinian
ring without zero divisors is a division
ring. (A ring has no zero divisors if
*ab*= 0 implies*a*= 0 or*b*= 0.) - 3.
- Show that there are no simple groups of order 300.
- 4.
- Let
*R*be a commutative domain with field of fractions*F*. Prove that*F*is an injective*R*-module. - 5.
- State and prove a structure theorem analogous to the
Fundamental Theorem for modules over a PID, which describes finitely
generated
/
*n*-modules. (You may assume the Fundamental Theorem for any argument.) - 6.
- Assume that
*K*/*F*is a Galois field extension and a lies in an algebraic closure of*K*. Prove that |Gal(*K*/*F*)| divides |Gal(*K*(a)/*F*(a))| deg a, where deg a denotes the degree of a over*F*. - 7.
- Prove that a finite
*p*-group with a unique subgroup of index*p*is cyclic. (Hint: first consider abelian*p*-groups.) - 8.
- Let
*f*:*M*- >*N*be a surjective homomorphism of left*R*-modules. Show that if*P*is a projective*R*-module, then the induced map*f*_{*}: Hom_{R}(*P*,*M*) - > Hom_{R}(*P*,*N*) is a surjection of abelian groups. - 9.
- Let
*k*be a field and let*R*=*k*[*X*_{1},...,*X*_{m}] be the polynomial ring in*m*indeterminates. Prove that if*M*is a simple*R*-module, then dim_{k}*M*< .