I got this question while I am studying Galois correspondence.
Let $K/F$ be an infinite extension and $G = \mathrm{Aut}(K/F)$. Let $H$ be a subgroup of $G$ with finite index and $K^H$ be the fixed field of $H$. Is it true that $[K^H:F]= (G:H)$?
For finite extension, I verified this is true. Is it true in infinite case also?
Thanks