Higher Algebra 2 — Spring 2006

Gerald Hoehn

Problem sheet 6 April 7, 2006

Problem 1: Let K/k be a Galois extension with Galois group G. Let H be a subgroup of G and denote the normalizer of H in G by N.

(a) Prove that N acts on K^{H} by automorphisms and that the field extension K^{H}/K^{N} is Galois with Galois group N/H.

(b) Prove that if φ ∈ Aut(K^{H}/k) then φ fixes K^{N}, i.e., K^{N} is the smallest subfield L of K^{H} containing k such that K^{H}/L is Galois.

Problem 2: Let K_{1} and K_{2} be two extension fields of a field k contained both in a field K, and assume that the two field extensions K_{1}/k and K_{2}/k are both Galois.

(a) Prove that K_{1} ∩ K_{2}/k is a Galois extension.

(b) Prove that K_{1}K_{2}/k is a Galois extension with Galois group

{(σ, τ) | σ|_{K_{1}∩K_{2}} = τ|_{K_{1}∩K_{2}}} < Aut(K_{1}/k) × Aut(K_{2}/k).

Problem 3: Let p be a prime and n be a natural number.

(a) Prove that there exists an irreducible polynomial of degree n over F_{p}.

(b) Prove that the product of all monic irreducible polynomials in F_{p}[x] of degree d where d runs through all the divisors of n equals x^{p^{n}} − x.

Problem 4:

(a) Find a primitive element for the field extension Q(\sqrt{2}, \sqrt{3}, \sqrt{5})/Q.

(b) Let L = F_{3}(x, y) be the field of rational functions in the variables x and y over the finite field F_{3} with three elements. Let K = F_{3}(x^{3}, y^{3}) ⊂ L. Prove that the field extension L/K is not simple. (Hint: Show that [L : K] = 9, but any simple extension of K by an element of L has degree at most 3.)