Is a Galois extension (reviewed)

Let $F$ be a subfield of $K$, $$ \Aut(K/F)=\{ \sigma:K\to K\mid \sigma(a)=a \text{ for all } a\in F \text{ and } \sigma \text{ is a ring homomorphism}\},$$ and $$ K^{\Aut(K/F)} = \{ a\in K \mid \sigma(a)=a\}.$$ Then $K$ is Galois over $F$ if $K^{\Aut(K/F)} = F$.