By David B Surowski

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56 CHAPTER 2. (f (x), f (x)) = 1, then f (x) has no repeated roots in any splitting field. ) (ii) If f (x) is irreducible, then f (x) is separable if and only if f (x) = 0. (iii) If F has characteristic p > 0, and if f (x) ∈ F[x] is irreducible but not separable, then f (x) = g(xp ) for some irreducible g(x) ∈ F[x]. Obviously, it follows that if char F = 0 then every polynomial f (x) ∈ F[x] is separable. 5 Let F ⊆ K be an algebraic extension of fields where F has characteristic p > 0. If α ∈ K, then α is separable over F if and only if F(α) = F(αp ).

9 If G is nilpotent, and if H is a proper subgroup of G, then H = NG (H). Thus, normalizers “grow” in nilpotent groups. 10 Let G be a finite group. Then G is nilpotent if and only if G is the direct product of its Sylow subgroups. 7 1. Show that H char N G⇒H G. 2. Let H be a subgroup of the group G with G ≤ H. Prove that H G. 3. Let G be a finite group and let P be a 2-Sylow subgroup of G. If M ≤ P is a subgroup of index 2 in P and if τ ∈ G is an involution not conjugate to any element of M , conclude that τ ∈ G (commutator subgroup).

4 Let E ⊇ F be an algebraic extension with F closed in E. Then every element of ΩE/F is closed in E. The field extension E ⊇ F is called a Galois extension (we sometimes say that E is Galois over F) if F is closed in E. Let E ⊇ F be a field extension with Galois group G, and let K ∈ ΩE/F . We say that K is stable if σK = K for each σ ∈ G. We denote by ΩcG the closed subgroups of G. 5 (Fundamental Theorem of Galois Theory) Let E ⊇ F be an algebraic Galois extension. 52 CHAPTER 2. FIELD AND GALOIS THEORY (i) The mappings Gal(E/•) : ΩE/F → ΩcG , inv• : ΩcG → ΩE/F are inverse isomorphisms.

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