Let $L=\mathbb{Q}(T_1,T_2,T_3,T_4)$ be a purely transcendental extension of $\mathbb{Q}$, and percieve $S_4$ on $\{T_1,T_2,T_3,T_4\}$ as subgroup of Aut$(L)$. Write $K=L^{S_4}$. Calculate $[L:K(\alpha)]$ for:
$\alpha=T_1+T_2+T_3+T_4$
$\alpha=T_1+T_2+T_3$
$\alpha=T_1+T_2$
$\alpha=T_1$
It seems to me that $\alpha=T_1+T_2+T_3+T_4$ is already in $K$ because it is invariant under all elements of $S_4$. But to me for all $\alpha$ the degree seems infinite, which must be wrong?