Definition 1

— Let E and F be two extensions of K contained in $\Omega$. We shall say that E and F are conjugate (in $\Omega$) if there exists a K-automorphism u of $\Omega$ such that $u(E) = F$. Two elements x and y of $\Omega$ are said to be conjugate over K if there exists a K-automorphism u of $\Omega$ such that $u(x) = y$.

Let E and F be two extensions of K contained in $\Omega$. By Prop. 1, E and F are conjugate over K if and only if they are isomorphic extensions of K. This is so in particular when there exist two subsets A and B of $\Omega$ such that $E = K(A)$ and $F = K(B)$ and a K-automorphism u of $\Omega$ such that $u(A) = B$.

The relation « x and y are conjugate over K » is an equivalence relation in $\Omega$; -the equivalence classes are called the conjugacy classes in $\Omega$; they are the orbits in $\Omega$ of the group of all K-automorphisms of $\Omega$.