Extensions of local fields and elementary symmetric polynomials

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 $\mathcal{M}_K$.