Doctoral Algebra Comprehensive Exam August 1994

1. Consider the symmetric group S_{4} on 4 elements.

(a) Find all Sylow subgroups of S_{4}.

(b) Give an example of a non-abelian group of order 24 which has a different number of p-Sylow subgroups than S_{4} has, for some p.

(c) Find a composition series for S_{4}.

2. Let R = k[x, y] be the polynomial ring in 2 variables over a field k. Consider the ideal I = (xy − x, y^{2} − y). Let S = R/I be the quotient ring.

(a) Show that m = (x, y) is a maximal ideal of R.

(b) Show that I ⊂ m.

(c) Let m̄ = m/I ⊂ S. Show that m̄ is a maximal ideal of S.

(d) Show the isomorphism of rings Sm̄ ≃ k, where Sm̄ denotes the localization of S at m̄.

3. Let R be a PID, and let F be a finitely generated free R-module. Prove that any submodule of F is also free. (You may not simply quote the theorem that states this fact.)

4. Consider the polynomial f(x) = x^{4} + x^{3} + x^{2} + 1 over the finite field F_{2} with two elements.

(a) Find the splitting field K of f(x).

(b) Show directly that this is a Galois extension, and find the Galois group of K over F_{2}.