Let $F \subset K$ be a finite extension of fields and let $G$ be the group of $F$-automorphisms of $K$. State a result relating the order of $G$ to the degree $[K: F]$.

Now let $K=k\left(X_{1}, \ldots, X_{4}\right)$ be the field of rational functions in four variables over a field $k$ and let $F=k\left(s_{1}, \ldots, s_{4}\right)$ where $s_{1}, \ldots, s_{4}$ are the elementary symmetric polynomials in $k\left[X_{1}, \ldots, X_{4}\right]$. Show that the degree $[K: F] \leqslant 4$ ! and deduce that $F$ is the fixed field of the natural action of the symmetric group $S_{4}$ on $K$.

Show that $X_{1} X_{3}+X_{2} X_{4}$ has a cubic minimum polynomial over $F$. Let $G=$ $\langle\sigma, \tau\rangle \subset S_{4}$ be the dihedral group generated by the permutations $\sigma=(1234)$ and $\tau=(13)$. Show that the fixed field of $G$ is $F\left(X_{1} X_{3}+X_{2} X_{4}\right)$. Find the fixed field of the subgroup $H=\left\langle\sigma^{2}, \tau\right\rangle$.