Perturbing Eisenstein polynomials over local fields

Kevin Keating

doi:10.5802/jtnb.1045

Kevin Keating ¹

¹ Department of Mathematics University of Florida Gainesville, FL 32611, USA

Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 681-694.

Abstract
Résumé

Let $K$ be a local field whose residue field has characteristic $p$ and let $L / K$ be a finite separable totally ramified extension. Let $π_{L}$ be a uniformizer for $L$ and let $f (X)$ be the minimum polynomial for $π_{L}$ over $K$ . Suppose ${\tilde{π}}_{L}$ is another uniformizer for $L$ such that ${\tilde{π}}_{L} \equiv π_{L} + r π_{L}^{ℓ + 1} (mod π_{L}^{ℓ + 2})$ for some $ℓ \geq 1$ and $r \in 𝒪_{K}$ . Let $\tilde{f} (X)$ be the minimum polynomial for ${\tilde{π}}_{L}$ over $K$ . In this paper we give congruences for the coefficients of $\tilde{f} (X)$ in terms of $ℓ$ , $r$ , and the coefficients of $f (X)$ . These congruences improve work of Krasner [8].

Soit $K$ un corps local de caractéristique résiduelle $p$ et soit $L / K$ une extension séparable finie totalement ramifiée. Soit $π_{L}$ une uniformisante de $L$ , de polynôme minimal $f (X)$ sur $K$ . Supposons que ${\tilde{π}}_{L}$ est une autre uniformisante de $L$ telle que ${\tilde{π}}_{L} \equiv π_{L} + r π_{L}^{ℓ + 1} (mod π_{L}^{ℓ + 2})$ pour certains $ℓ \geq 1$ et $r \in 𝒪_{K}$ . Soit $\tilde{f} (X)$ le polynôme minimal de ${\tilde{π}}_{L}$ sur $K$ . Dans cet article nous donnons des congruences pour les coefficients de $\tilde{f} (X)$ en termes de $ℓ$ , $r$ , et les coefficients de $f (X)$ . Ces congruences améliorent le travail de Krasner [8].

Received: 2017-01-07
Revised: 2017-03-05
Accepted: 2017-06-15
Published online: 2018-12-06

MR: 3891333 | Zbl: 1441.11294

DOI: 10.5802/jtnb.1045

Classification: 11S15, 11S05
Keywords: local fields, Eisenstein polynomials, symmetric polynomials, indices of inseparability, digraphs

Author's affiliations:

Kevin Keating ¹

¹ Department of Mathematics University of Florida Gainesville, FL 32611, USA

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Kevin Keating. Perturbing Eisenstein polynomials over local fields. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 681-694. doi : 10.5802/jtnb.1045. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1045/

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