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 se...