Algebra Prelim, August 1999

Answer all questions

  1. Let G be a simple group of order 480 with an abelian Sylow 2-subgroup.

    1. If P and Q are distinct Sylow 2-subgroups of G, by considering CG(P $ \cap$ Q), prove that P $ \cap$ Q = 1.

    2. Prove that there is no such group G.

  2. Let R be a UFD, let S be a multiplicatively closed subset of R such that 0$ \notin$S, and let p be a prime in R. Prove that p/1 is either a prime or a unit in S-1R.

  3. Let k be the field $ \mathbb {Z}$/2$ \mathbb {Z}$. Classify the finitely generated projective k[X]/(X3 + X)-modules up to isomorphism.

  4. Let R be a ring, let M be a Noetherian R-module, and let J denote the Jacobson radical of R. Prove that either MJn = 0 for some positive integer n, or MJn + 1 $ \subset$ MJn (strict inequality) for all positive integers n.

  5. Let R be a nonzero right Artinian ring (with a 1) with no nonzero nilpotent ideals and no nontrivial ($ \ne$0, 1) idempotents. Prove that R is a division ring.

  6. Compute the character table of S4.

  7. Let K be a splitting field of the polynomial X4 - 2 over $ \mathbb {Q}$. Determine the order of Gal(K/$ \mathbb {Q}$). Use this to show that K contains a subfield L such that [L : $ \mathbb {Q}$] = 4 and L is normal over $ \mathbb {Q}$.

Peter Linnell