I was a reading a proof in Lang, and I think he seems to be using the following:
Let $k$ be a field and $E$ and algebraic extension lying within a given algebraic closure $k^a$ of $k$. Suppose, there is an embedding $\sigma$ of $E$ into $k^a$ that fixes $k$ s.t. $\sigma(E)\subseteq E$ then, $\sigma(E)=E$.
I am generalizing this from a specific statement, so I could be wrong, but I am unable to prove this when the extension $E/k$ is not finite. In the finite case, I can just compare the degrees of the extensions.