I hope this isn't too elementary of a question, but I'm not sure I understand Artin's proof that if $K/F$ is a finite extension, then $K/F$ Galois is equivalent to $K$ being a splitting field over $F$ (this is Theorem 16.6.4 in the second edition). We're working in characteristic zero here. I understand one direction, that Galois implies splitting field: if we let $\gamma_{1}$ generate $K/F$ and $f$ be its minimal polynomial of degree $n$, each automorphism of $K/F$ comes from sending $\gamma_{1}$ to another root of $f$, and in order for $\operatorname{Gal}(K/F)$ to have order $n$ we need all of the roots of $f$ to be in $K$.
However, I still am not seeing the other direction - after stating the above Artin seems to finish by saying that if we define $\gamma_{1},f$ as before, "$K$ is a splitting field over $F$ iff $f$ splits completely in $K$." But isn't it possible that $f$ doesn't split, but $K$ is the splitting field of some other polynomial?