Example 1

Suppose that the characteristic of K is different from 2 and let N be a quadratic extension of K. If x ∈ N − K, then we have N = K(x) and the minimal polynomial of x over K is of the form f(X) = X² − aX + b, with a, b ∈ K. We thus have f(X) = (X − x)(X − y) where y = a − x, so y is conjugate to x; since f(X) is separable, the extension N is Galois. The group Gal(N/K) has two elements which induce the two permutations of the set {x, y}.