Theorem 2 (Steinitz)

— Let $K$ be a field; then there exists an algebraic closure of $K$. If $\Omega$ and $\Omega'$ are two algebraic closures of $K$, there exists a $K$-isomorphism of $\Omega$ onto $\Omega'$.

By Prop. 6 an algebraic closure of $K$ is nothing other than a splitting extension for the set of all non-constant polynomials in $K[X]$. Th. 2 therefore follows from V, p. 21, Prop. 4 and V, p. 22, Cor.