Group superscript on field

Question

If $N$ is a collection of maps $K \rightarrow K$ that fix F, and $K/F$ is a (Galois) field extension, what does $K^N$ denote?

For context, I've seen this in algebraic number theory textbooks. Does this require the Galois assumption?

Answer 1

It consists of the elements of $K$ that are fixed by all the maps in $N$, which gives us a subfield of $K$. This certainly contains $F$ and therefore is an intermediate extension $K/K^N/F$, but can often contain more than just $F$. There is no need for a Galois assumption here.

When $K/F$ is finite and Galois, there exists an inclusion-reversing bijection between subgroups $H$ of the Galois group $G = Gal(K/F)$, and intermediate field extensions $K/K'/F$. This bijection is given explicitly by $H \mapsto K^H$ and its inverse is given by $K' \mapsto Gal(K/K') \subset Gal(K/F)$. This is the fundamental theorem of Galois theory.