Algebra Prelim, Winter 2003

Instructions: do all problems.

  1. Let K denote the splitting field over the rational numbers Q of the polynomial f (x) = x5 + x4 + 3x + 3.
    What is [K : Q]?

    Determine the Galois group Gal(K/Q).

  2. Prove or disprove: If x and y are elements of a finite abelian group G with the same order, then there is an automorphism q of G such that q(x) = y.

  3. Let H be a group of order 2002. Prove that the number of elements in the set {h $ \in$ H | h2 = e} is even (where e is the identity of H).

  4. Let G be a group with the following property: for each g $ \in$ G - 1, there exists a normal subgroup K of G such that G/K is abelian and g$ \notin$K (where 1 is the trivial subgroup). Prove that G is abelian.

  5. Prove that if A and B are commutative Noetherian rings, then so is the cartesian product A X B.

  6. Let R be a commutative ring, let P be a projective R-module and let I be an ideal in R. Prove P/IP is a projective R/I-module.

  7. Let k be a field and let I be an ideal of k[x] (where k[x] is the polynomial ring over k in the variable x).
    Show that k + I is a subring of k[x].

    Prove that if I $ \neq$ 0 then k[x] $ \otimes_{k+I}^{}$ k[x]/I is finite dimensional over k.

Peter Linnell