Higher Algebra 2 — Spring 2006

Gerald Hoehn

Problem sheet 5 March 17, 2006

Problem 1: (a) Let K be a field of characteristic p > 0. Let L be a finite extension of K and suppose that [L : K] is prime to p. Show that L is separable over K.

(b) Let K/k be a separable extension, let L/k be an arbitrary extension and assume K and L are subfields of a common extension field. Prove that the field extension KL/L is separable.

Problem 2: Let K/k be an algebraic field extension, where k is of characteristic p > 0. An element a ∈ K is called purely inseparable over k, if its minimal polynomial has only one different root. Prove that the following statements are equivalent:

1. The extension K/k is purely inseparable, i.e., every element a ∈ K is purely inseparable over k.

2. The extension K/k has separable degree [K : k]_{s} = 1.

3. For every element a ∈ K there exists an n ∈ N ∪ {0} such that a^{p^{n}} ∈ k.

4. The minimal polynomial of every element a ∈ K is of the form x^{p^{n}} − b, where b ∈ k and n ∈ N ∪ {0}.

5. There exists a set of generators of K over k which are purely inseperable over k.

Problem 3: Determine the splitting field of f(x) = x^{4} − 4 x^{2} − 1 ∈ Q[x] in C, determine its Galois group over Q, and describe the lattices of its subfields and of subgroups of the Galois group.

Problem 4: Show that the Galois group of f(x) = x^{5} − 4 x + 2 ∈ Q[x] over Q is the symmetric group S_{5}.