Theorem 1 (Steinitz)

Let K be a field, E an algebraic extension of K and $\Omega$ an algebraically closed extension of K; then there exists a K-homomorphism of E into R.

By V, p. 13, Scholium, there exists an extension field $\Omega'$ of $\Omega$ and a K-homomorphism u of E into $\Omega'$. Let $x \in E$; since x is algebraic over K, u(x) is algebraic over u(K) and a fortiori over $\Omega$ (V, p. 17, Cor. 2); since $\Omega$ is algebraically closed, we thus have $u(x) \in O$. It follows that $u$ maps $E$ into $O$.