Characteristic Polynomial of Galois automorphism

Question

Let $K/F$ be a finite Galois extension.

Let $g$ be an element of $Gal(K/F)$

How do I compute the characteristic polynomial of $g$, where $g$ is considered as a $F$-linear map $K \rightarrow K$?

Answer 1

A Galois extension $K/F$ with Galois group $g$ has a normal basis: that means, there is an $a\in K$ such that $B=\{g(a):g\in G\}$ is an $F$-basis of $K$. If $h$ is in $G$ then the matrix of $h$ with respect to the basis $B$ is easy to describe. Can you do that?

Answer 2

We have $K\cong F[G]$ as an $F[G]$-module; this is the content of the normal basis theorem. So it suffices to consider the action of $g$ on the space $F[G]$. Observe that, with the basis $G$, $g$ essentially acts as a permutation matrix, associated to the permutation $g$ acts as on $G$ itself.

The characteristic polynomial of a permutation matrix depends only on the cycle structure of the permutation (which makes sense, since cycle types enumerate conjugacy classes and characteristic polynomials are conjugacy-invariant).