Vol. 264, No. 1, 2013

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On a Galois connection between the subfield lattice and the multiplicative subgroup lattice

John K. McVey

Vol. 264 (2013), No. 1, 213–219

Given finite fields F < E, we present a collection of subgroups C E× and establish, to each C, a Galois connection between the intermediate field lattice = {LF L E} and C’s subgroup lattice. Our main result is that, in all but an extremely limited and completely determined family, the closed subset of is itself, establishing a natural bijection between and the lattice {L CL ∈ℰ}. As an application, we use this bijection to calculate the set of degrees for the complex-valued irreducible representations of the split extension C Gal(E∕F).

Galois correspondence, lattice, character degree, finite field
Mathematical Subject Classification 2010
Primary: 06A15
Secondary: 20C15
Received: 2 June 2012
Revised: 10 December 2012
Accepted: 26 December 2012
Published: 5 July 2013
John K. McVey
Department of Mathematics
Kent State University
Mathematics and Computer Science Building 233
Summit Street
Kent, OH 44242
United States