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

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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.

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