Computing Character Degrees via a Galois Connection

In a previous paper, the second author established that, given finite fields F < E and certain subgroups C ≤ E×, there is a Galois connection between the intermediate field lattice {L | F ≤ L ≤ E} and C’s subgroup lattice. Based on the Galois connection, the paper then calculated the irreducible, complex character degrees of the semi-direct product C⋊Gal(E/F ). However, the analysis when |F | is a Mersenne prime is more complicated, so certain cases were omitted from that paper. The present exposition, which is a reworking of the previous article, provides a uniform analysis over all the families, including the previously undetermined ones. In the group C ⋊Gal(E/F), we use the Galois connection to calculate stabilizers of linear characters, and these stabilizers determine the full character degree set. This is shown for each subgroup C ≤ E× which satisfies the condition that every prime dividing |E× : C| divides |F×|.