In this work in progress I state the following Fundamental Theorem of Galois Theory:
Let $K/F$ be any field extension. The following are equivalent:
(i) For all subextensions $L$ of $K/F$, $K^{\operatorname{Aut}(K/L)} = L$.
(ii) At least one of the following holds:
(a) $K/F$ is algebraic, normal and separable (i.e., a Galois extension in the usual sense), or
(b) $K$ is algebraically closed of characteristic zero.
That's the good news. The bad news is that this "Theorem", which I wrote down several years ago, is actually only a conjecture (it has remained as an open question on MathOverflow for more than a year, which is some indication of its nontriviality, at least). Okay, but there is more good news: the implication (ii) $\implies$ (i) is proven, and is a fairly routine application of basic field theory. Your question is a special case of (ii) $\implies$ (i), so there you go.
Added: Mea culpa, the proof of (ii) $\implies$ (i) in the linked to notes is "$\ldots$". (When you can't prove the big theorem you announce on the first page, you lose some motivation to fill in the other details, it seems.) Instead, please see $\S 10.1$ of my field theory notes in which there is a complete proof of (even a result slightly more general than) (ii) $\implies$ (i). Really, I promise.