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 C_G(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 MJ^{n + 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 $\mathbb {Q}$.