Algebra 819 - Homework 4 Samuel Otten

Let K ≤ F be a normal field extension. Put S := S(K,F) and P := P(K,F). Let K ≤ E ≤ S. (a) We will show that S(E,F) = S. Let a ∈ S. Then a is separable over K. By 5.2.20, a is also separable over E, so a ∈ S(E,F) and S ⊆ S(E,F). Now note that E is a separable extension of K because E ≤ S. Let a ∈ S(E,F). Since a is separable over E, E(a) is a separable extension of E. Thus, K ≤ E ≤ E(a) is a sequence of separable extensions. This means K ≤ E(a) is a separable extension (5.2.25) and so a ∈ S. (b) We will show that EP ≤ F is separable. Note that P ≤ F is separable because K ≤ F is normal (5.2.24). And E ≤ F is separable because E is a subset of the separable elements over K. We know K = E ∩ P and so any element ep ∈ EP is in K and so separable over K or is a root of me m K p . The polynomial m K p only contributes one root, namely p. If p is also a root of m K e then me is the minimal polynomial of ep which is separable. If p is not a root of m K e then m K e m K p is the minimal polymomial of ep which is also separable. So EP ≤ F is a separable extension.