Definition 1

— An extension N of K is said to be Galois if it is algebraic and satisfies the equivalent conditions a), b), c) of Th. 1.

Let N be a field, Γ a group of automorphisms of N and $N_0$ the field of invariants of Γ. When N is algebraic over $N_0$ it is a Galois extension of $N_0$. This need not always be so: for example, suppose that K is infinite and take N to be the field of rational fractions K(X); for each $a \in K$ let $\sigma_a$ be the automorphism of K(X) which maps $f(X)$ to $f(X + a)$. The set of all $a$, is a group of automorphisms of K(X) whose field of invariants is easily seen to be K; however K(X) is not algebraic over K.

Let Ω be an algebraic closure of K, let A denote a set of elements of Ω separable over K and B the set of conjugates over K of elements of A. Then B consists of elements that are algebraic and separable over K. Therefore (V, p. 39, Prop. 6 and p. 56, Prop. 5) the field K(B) is a separable quasi-Galois extension of K; in other words, the quasi-Galois extension generated by A (V, p. 55) is a Galois extension of K; we shall also say that it is the Galois extension of K generated by the subset A of O.

In particular, the splitting field in Ω of a family of separable polynomials over K, a separable closure of K, are Galois extensions of K.