Algebra Prelim, January 1998

1. 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.
1. Describe the factorization of f (x) into irreducible polynomials in F[x].

2. Describe the factorization of f (x) into irreducible polynomials in K[x].

2. Let G be a group of order 1947 = 3*11*59. Prove that G is cyclic.

3. Let G be a group of order pn with n2 and p prime. Prove that G has a normal abelian subgroup of order p2.

4. Let K = (,w) where w = cos(2p/3) + isin(2p/3) is a primitive cube root of unity.
1. What is [K : ]?

2. Prove that K/ is a Galois extension.

3. Describe the Galois group of K/.

5. 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.
1. Prove that A R is an ideal of R.

2. If A R = Rd, prove that A = Sd. (Hint: if a/b S with (a, b) = 1, prove that 1/b S.)

6. 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 pak. Let M be a normal subgroup of G and let P be a Sylow p-subgroup of G.
1. Prove that PM/M is a Sylow p-subgroup of G/M.

2. 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.

3. Prove that the number of Sylow p-subgroups of G/M is a divisor of the number of Sylow p-subgroups of G.

1. Give an example of a group of order 3540 = 59*60 = 59*5*3*4 which is not solvable.

2. Give an example of a group of order 3540 which is solvable but not cyclic.

7. Let G and H be finitely generated abelian groups such that G G H H. Prove that G H.

Peter Linnell
1999-06-16