Look at the following proposition:
Let $K\subset L\subset M$ be three fields. If $M$ is a splitting field over $K$ of a polinomial in $K[X]$ and moreover if for every $\sigma\in G=Gal(M/K)$ we have that $\sigma(L)\subseteq L$, then $L$ is a splitting field over $K$.
My attempt of proof is the following, but I have problems in the last part:
if $M=K(a_1,\ldots, a_n)$ where $a_1,a_2,\ldots, a_n$ are the roots of $f\in K[X]$, then (rilabelling the roots if it is necessary) we have that $L=K(a_1,\ldots,a_t)$ with $t<n$. Now if $g=(X-a_1)\ldots(X-a_n)$, by the property of $L$, we can argue that $g\in Fix(G)[X]$, and so $L$ is the splitting field of $g$ over $Fix(G)[X]$ but not over $K$. In characteristic $0$ we could conclude that $Fix(G)=K$, but what about the general case?