Proposition 7

— Let E be a field, G a group of automorphisms of E and K the subfield of E consisting of the invariants of G. Then E is a separable extension of K.

Let L be an extension of K; there exists an algebraically closed extension Ω of L whose transcendence degree over K is at least equal to that of E over K. By Cor. 1 (V, p. 117) there exists a K-homomorphism u of E into R. We denote by v the Ω-algebra homomorphism of A = Ω ⊗_K E into Ω which maps λ ⊗ x to λ . u(x) for λ ∈ Ω and x ∈ Ω; we write a for the kernel of v.

For every s ∈ G let h_s be the automorphism Id, ⊗ s of the 0-algebra A; the kernel of the homomorphism v ∘ h_s of A into Ω is the prime ideal a_s = h_s^{-1}(a) of A. It is clear that the ideal b = ⋂_{s ∈ G} a_s of A is stable under the automorphisms h_s. Therefore (V, p. 63, Cor.) the ideal b is of the form c ⊗_K E, where c is an ideal of Ω. Now b ⊂ a ≠ A, and so c ≠ Ω; since Ω is a field, we have c = 0, hence b = 0.

The family (a_s), s ∈ G of prime ideals of A thus has zero intersection. By Prop. 2 (V, p. 118) the ring is reduced, and a fortiori the same holds of the subring L ⊗K E of A. Since L was an arbitrary extension of K, this proves that E is separable over K.