Algebra Prelim, January 1998

Answer all questions

  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 n$ \ge$2 and p prime. Prove that G has a normal abelian subgroup of order p2.

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

    2. Prove that K/$ \mathbb {Q}$ is a Galois extension.

    3. Describe the Galois group of K/$ \mathbb {Q}$.

  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 $ \cap$ R is an ideal of R.

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

  8. Let G and H be finitely generated abelian groups such that G $ \oplus$ G $ \cong$ H $ \oplus$ H. Prove that G $ \cong$ H.



Peter Linnell
1999-06-16