Proposition 1

— Let E be an extension of K contained in $\Omega$ and u a K-homomorphism of E into $\Omega$.

a) If u maps E into E, u induces a K-automorphism of E.

b) There exists a K-automorphism v of $\Omega$ extending u.

Suppose that $u(E) \subset E$; to prove a) it is enough to show that $u(E) = E$. Let x be an element of E, f the minimal polynomial of x over K and $\Phi$ the set of roots of fin E. The set $\Phi$ is finite, the mapping u of E into E is injective and we have $u(\Phi) \subset \Phi$; therefore $u(\Phi) = \Phi$, whence $x \in u(\Phi) \subset u(E)$; this shows that $E = u(E)$.

It is clear that $\Omega$ is an algebraic closure of E and of $u(E)$; therefore (V, p. 23, Cor.), the isomorphism u of E onto $u(E)$ extends to an isomorphism v of $\Omega$ onto $\Omega$.