The first digit of the discriminant of Eisenstein polynomials as an invariant of totally ramified extensions of p-adic fields

Awtrey, Chad; Gaura, Alexander; Pauli, Sebastian; Rudzinski, Sandi; Uy, Ariel; Zinzer, Scott

doi:10.2140/involve.2020.13.747

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-4184 (online)

ISSN 1944-4176 (print)

Author index

To appear

Other MSP journals

This article is available for purchase or by subscription. See below.

The first digit of the discriminant of Eisenstein polynomials as an invariant of totally ramified extensions of $p$-adic fields

Chad Awtrey, Alexander Gaura, Sebastian Pauli, Sandi Rudzinski, Ariel Uy and Scott Zinzer

Vol. 13 (2020), No. 5, 747–758

DOI: 10.2140/involve.2020.13.747

Abstract

Let $K$ be an extension of the $p$ -adic numbers with uniformizer $π$ . Let $φ$ and $ψ$ be Eisenstein polynomials over $K$ of degree $n$ that generate isomorphic extensions. We show that if the cardinality of the residue class field of $K$ divides $n (n - 1)$ , then $v (disc (φ) - disc (ψ)) > v (disc (φ))$ . This makes the first (nonzero) digit of the $π$ -adic expansion of $disc (φ)$ an invariant of the extension generated by $φ$ . Furthermore we find that noncyclic extensions of degree $p$ of the field of $p$ -adic numbers are uniquely determined by this invariant.

PDF Access Denied

We have not been able to recognize your IP address 18.97.14.86 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 30.00:

Keywords

$p$-adic field, Eisenstein polynomial, discriminant, invariant

Mathematical Subject Classification

Primary: 11S05, 11S15

Milestones

Received: 22 April 2019

Revised: 17 February 2020

Accepted: 15 September 2020

Published: 5 December 2020

Communicated by Kenneth S. Berenhaut

Authors

Chad Awtrey
Department of Mathematics and Statistics Elon University Elon, NC United States

Alexander Gaura
Department of Mathematics Princeton University Princeton, NJ United States

Sebastian Pauli
Department of Mathematics and Statistics University of North Carolina Greensboro, NC United States

Sandi Rudzinski
Department of Mathematics and Statistics University of North Carolina Greensboro, NC United States

Ariel Uy
Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA United States

Scott Zinzer
Department of Mathematics Aurora University Aurora, IL United States

Vol. 13, No. 5, 2020