Extensions of local fields and elementary symmetric polynomials

Kevin Keating

doi:10.5802/jtnb.1032

Kevin Keating ¹

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

Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 431-448

Abstract (Original language)
Résumé

Let $K$ be a local field whose residue field has characteristic $p$ and let $L / K$ be a finite separable totally ramified extension of degree $n$ . Let $σ_{1}, ..., σ_{n}$ denote the $K$ -embeddings of $L$ into a separable closure of $K$ . For $1 \leq h \leq n$ let $e_{h} (X_{1}, ..., X_{n})$ denote the $h$ th elementary symmetric polynomial in $n$ variables, and for $α \in L$ set $E_{h} (α) = e_{h} (σ_{1} (α), ..., σ_{n} (α))$ . Let $𝒫_{K}$ be the maximal ideal of the ring of integers of $K$ and let $j = min {v_{p} (h), v_{p} (n)}$ . We show that for $r \in ℤ$ we have $E_{h} (𝒫_{L}^{r}) \subset 𝒫_{K}^{⌈ (i_{j} + h r) / n ⌉}$ , where $i_{j}$ is the $j$ th index of inseparability of $L / K$ . In certain cases we also show that $E_{h} (𝒫_{L}^{r})$ is not contained in any higher power of $𝒫_{K}$ .

Soit $K$ un corps local de caractéristique résiduelle $p$ et soit $L / K$ une extension séparable finie totalement ramifiée de degré $n$ . Soient $σ_{1}, ..., σ_{n}$ les $K$ -plongements de $L$ dans une clôture séparable de $K$ . Pour tout $1 \leq h \leq n$ , soit $e_{h} (X_{1}, ..., X_{n})$ le polynôme symétrique élémentaire en $n$ variables de degré $h$ , et pour tout $α \in L$ , soit $E_{h} (α) = E_{h} (σ_{1} (α), ..., σ_{n} (α))$ . Soit $𝒫_{K}$ l’idéal maximal de l’anneau des entiers de $K$ et soit $j = min {v_{p} (h), v_{p} (n)}$ . Nous montrons que $E_{h} (𝒫_{L}^{r}) \subset 𝒫_{K}^{⌈ (i_{j} + h r) / n ⌉}$ pour tout $r \in ℤ$ , où $i_{j}$ est l’indice d’inséparabilité d’ordre $j$ de l’extension $L / K$ . Dans certains cas, nous montrons également que $E_{h} (𝒫_{L}^{r})$ n’est contenu dans aucune puissance supérieure de $𝒫_{K}$ .

Article information
Cite this article

Received: 2016-08-25
Revised: 2017-06-08
Accepted: 2017-06-15
Published online: 2018-12-06

MR Zbl

DOI: 10.5802/jtnb.1032

Classification: 11S15, 11S05
Keywords: local fields, symmetric polynomials, norm, trace, 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

Kevin Keating. Extensions of local fields and elementary symmetric polynomials. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 431-448. doi: 10.5802/jtnb.1032

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