B4.3

Galois Theory
Part II, 2001

Define the concept of separability and normality for algebraic field extensions. Suppose K=k(α)K=k(\alpha) is a simple algebraic extension of kk, and that Aut(K/k)\operatorname{Aut}(K / k) denotes the group of kk-automorphisms of KK. Prove that Aut(K/k)[K:k]|\operatorname{Aut}(K / k)| \leqslant[K: k], with equality if and only if K/kK / k is normal and separable.

[You may assume that the splitting field of a separable polynomial fk[X]f \in k[X] is normal and separable over kk.]

Suppose now that GG is a finite group of automorphisms of a field FF, and E=FGE=F^{G} is the fixed subfield. Prove:

(i) F/EF / E is separable.

(ii) G=Aut(F/E)G=\operatorname{Aut}(F / E) and [F:E]=G[F: E]=|G|.

(iii) F/EF / E is normal.

[The Primitive Element Theorem for finite separable extensions may be used without proof.]