Vol. 14, No. 3, 2021

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Eisenstein polynomials defining Galois dihedral $p$-adic fields of degree $2p$

Chad Awtrey, Nicholas Hadgis and Annalise Von Sprecken

Vol. 14 (2021), No. 3, 463–474
Abstract

Let p > 2 be prime, p the field of p-adic numbers, and f(x) p[x] an Eisenstein polynomial of degree 2p. We give necessary and sufficient conditions on the coefficients of f(x) for its Galois group to be isomorphic to the dihedral group of order 2p.

Keywords
Eisenstein polynomials, $p$-adic fields, dihedral, totally ramified extensions
Mathematical Subject Classification
Primary: 11S05, 11S15, 11S20
Milestones
Received: 20 September 2020
Revised: 2 February 2021
Accepted: 12 February 2021
Published: 17 July 2021

Communicated by Kenneth S. Berenhaut
Authors
Chad Awtrey
Department of Mathematics and Computer Science
Samford University
Homewood, AL
United States
Nicholas Hadgis
Department of Physical Therapy
Duke University School of Medicine
Durham, NC
United States
Annalise Von Sprecken
Department of Mathematics and Statistics
Elon University
Elon, NC
United States