Permutation of a fixed field is an intermediate field corresponding with the conjugate of the group corresponding to the fixed field

Question

The following is my question: Let $K/F$ be a Galois extension with Galois group $G = Gal(K/F)$, with intermediate field $L: F \subseteq L \subseteq K$ which corresponds to subgroup $H \leq G$ by the fundamental theorem of Galois theory (FTGT); $L: H = Gal(K/L), L = K^H$, where $K^H$ is the subfield of $K$ fixed by $H$. Let $\sigma \in G$. Prove that $\sigma(L)$ is the intermediate field which corresponds with the subgroup $\sigma H \sigma^{-1} \leq G$ by the FTGT.

I am expecting normalcy to come up but beyond that intuition, I am stumped.

Answer 1

An element $x$ of $K$ is fixed by $\sigma H\sigma^{-1}$ iff, for all $h\in H$, we have $\sigma (h(\sigma^{-1}(x)))=x$, which is equivalent to $h(\sigma^{-1}(x))=\sigma^{-1}(x)$. That last equation says that $\sigma^{-1}(x)$ is fixed by $h$, so the equation holds for all $h\in H$ iff $\sigma^{-1}(x)\in L$, i.e., iff $x\in\sigma(L)$.