Extensions of local fields and elementary symmetric polynomials

TL;DR

This paper investigates how elementary symmetric polynomials of elements in a ramified extension of local fields relate to the valuation structure, revealing inclusion relations involving the indices of inseparability.

Contribution

It extends the understanding of elementary symmetric polynomials in local field extensions by establishing valuation bounds linked to inseparability indices.

Findings

E_h( ext{elements in } ext{maximal ideal}^r) ext{ is contained in } ext{maximal ideal}^{ ext{ceil}((i_j+hr)/n)}

In certain cases, E_h( ext{maximal ideal}^r) is not contained in any higher power of the maximal ideal

Provides explicit valuation bounds connecting elementary symmetric polynomials and inseparability indices

Abstract

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 $n=up^{\nu}$. Let $\sigma_1,\dots,\sigma_n$ denote the $K$-embeddings of $L$ into a separable closure $K^{sep}$ of $K$. For $1\le h\le n$ let $e_h(X_1,\dots,X_n)$ denote the $h$th elementary symmetric polynomial in $n$ variables, and for $\alpha\in L$ set $E_h(\alpha) =e_h(\sigma_1(\alpha),\dots,\sigma_n(\alpha))$. Set $j=\min\{v_p(h),\nu\}$. We show that for $r\in\mathbb{Z}$ we have $E_h(\mathcal{M}_L^r)\subset \mathcal{M}_K^{\lceil(i_j+hr)/n\rceil}$, where $i_j$ is the $j$th index of inseparability of $L/K$. In certain cases we also show that $E_h(\mathcal{M}_L^r)$ is not contained in any higher power of $\mathcal{M}_K$.