Answer all questions

- Let
*F*and*K*be fields of characteristic 0 with*K*an extension of*F*of degree 21. Let*f*(*x*) be a polynomial in*F*[*x*] of degree 6 which has no roots in*F*and exactly two roots in*K*.- Describe the factorization of
*f*(*x*) into irreducible polynomials in*F*[*x*]. - Describe the factorization of
*f*(*x*) into irreducible polynomials in*K*[*x*].

- Describe the factorization of
- Let
*G*be a group of order 1947 = 3*11*59. Prove that*G*is cyclic. - Let
*G*be a group of order*p*^{n}with*n*2 and*p*prime. Prove that*G*has a normal abelian subgroup of order*p*^{2}. - Let
*K*= (,w) where w = cos(2p/3) +*i*sin(2p/3) is a primitive cube root of unity.- What is
[
*K*: ]? - Prove that
*K*/ is a Galois extension. - Describe the Galois group of
*K*/.

- What is
[
- Let
*R*be a PID with field of fractions (quotient field)*F*, let*S*be subring of*F*which contains*R*, and let*A*be an ideal of*S*.- Prove that
*A**R*is an ideal of*R*. - If
*A**R*=*Rd*, prove that*A*=*Sd*. (Hint: if*a*/*b**S*with (*a*,*b*) = 1, prove that 1/*b**S*.)

- Prove that
- Let
*p*be a prime, let*a*,*k*be positive integers such that*p*does not divide*k*, and let*G*be a group of order*p*^{a}*k*. Let*M*be a normal subgroup of*G*and let*P*be a Sylow*p*-subgroup of*G*.- Prove that
*PM*/*M*is a Sylow*p*-subgroup of*G*/*M*. - Let
*H*/*M*be the normalizer of*PM*/*M*in*G*/*M*and let*N*be the normalizer of*P*in*G*. Prove that*N**H*. - Prove that the number of Sylow
*p*-subgroups of*G*/*M*is a divisor of the number of Sylow*p*-subgroups of*G*.

- Prove that
- Give an example of a group of order
3540 = 59*60 = 59*5*3*4
which is not solvable.
- Give an example of a group of order 3540 which is solvable but not cyclic.

- Give an example of a group of order
3540 = 59*60 = 59*5*3*4
which is not solvable.
- Let
*G*and*H*be finitely generated abelian groups such that*G**G**H**H*. Prove that*G**H*.