I am trying to understand this problem. Part of it says
Let $K/F$ be any finite extension and let $\alpha \in K$. Let $L$ be a Galois extension of $F$ containing $K$ and let $H \leq Gal(L/F)$ be the subgroup corresponding to $K$. Define the norm of $\alpha$ from $K$ to $F$ to be $N_{K/F}(\alpha) = \prod_\sigma \sigma(\alpha)$, where the product is taken over all the embeddings of $K$ into an algebraic closure of $F$ (so over a set of coset representatives for $H$ in $Gal(L/F)$ by the Fundamental Theorem of Galois Theory). This is a product of Galois conjugates of $\alpha$. In particular, if $K/F$ is Galois this is $\prod_{\sigma \in Gal(K/F)} \sigma(\alpha)$.
(1) In the Fundamental Theorem of Galois Theory they say "the isomorphisms of $K$ (where $F \leq K \leq L$ and $L$ is Galois over $F$) into a fixed algebraic closure of $F$ containing $L$) which fix $F$ are in one-to-one correspondence with the cosets $\{\sigma H\}$ of $H$ in $G$." Are these isomorphisms part of the automorphism group of $L$?
(2) I am confused about the last sentence. They say you are taking the product over $\sigma \in Gal(K/F)$. However, aren't those all part of the same coset, namely $H$?
Thanks!