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

Awtrey, Chad; Hadgis, Nicholas; Von Sprecken, Annalise

doi:10.2140/involve.2021.14.463

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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

DOI: 10.2140/involve.2021.14.463

Abstract

Let $p > 2$ be prime, $ℚ_{p}$ the field of $p$ -adic numbers, and $f (x) \in ℚ_{p} [x]$ an Eisenstein polynomial of degree $2 p$ . 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 $2 p$ .

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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

Vol. 14, No. 3, 2021