What does it mean to say that a field is algebraically closed? Show that a field $M$ is algebraically closed if and only if, for any finite extension $L / K$ and every homomorphism $\sigma: K \hookrightarrow M$, there exists a homomorphism $L \hookrightarrow M$ whose restriction to $K$ is $\sigma$.

Let $K$ be a field of characteristic zero, and $M / K$ an algebraic extension such that every nonconstant polynomial over $K$ has at least one root in $M$. Prove that $M$ is algebraically closed.