Using a Galois connection to compute character degrees

Abstract

Given a Mersenne prime $q$ and a positive even integer $e$, let $F$ and $E$ be the fields of orders $q$ and $q^e$ respectively. Let $C$ be a cyclic subgroup of $E^{\times }$ whose index in $E^{\times }$ is divisible only by primes dividing $q-1$. We compute the character degrees of the group $C⋊\mathrm{Gal}(E/F)$ by using the Galois connection between the subfields of $E$ and the Galois group $\mathrm{Gal}(E/F)$.