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 \leq E^\times‎, ‎there is a Galois connection between the intermediate field lattice \{L \mid F \leq L \leq 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 \rtimes {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\rtimes{\rm 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\leq E^\times which satisfies the condition that every prime dividing |E^\times‎ :‎C| divides |F^\times|.

Keywords:

Galois correspondence , lattice , character degree , finite field