Proposition 8

— Let $\Omega$ be an algebraically closed extension of a field $K$, $E$ and $F$ two subextensions of $\Omega$ and $u$ a $K$-isomorphism of $E$ onto $F$. For a $K$-automorphism $v$ of $O$ to exist which extends $u$, it is necessary and sufficient for $\Omega$ to have the same transcendence degree over $E$ and $F$.

The condition is clearly necessary.

Suppose then that $\Omega$ has the same transcendence degree over $E$ and $F$ and let us choose a transcendence basis $B$ of $\Omega$ over $E$ and a transcendence basis $C$ of $\Omega$ over $F$. Since $B$ and $C$ are equipotent, Prop. 2 ($V$, p. 106) shows that $u$ extends to a $K$-isomorphism $u'$ of $E(B)$ onto $F(C)$. Since $C?$ is an algebraic closure of $E(B)$ and $F(C)$, the Cor. of $V$, p. 23 shows that $u'$ extends to an automorphism $v$ of $\Omega$.