Extensions of local fields and elementary symmetric polynomials

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 $n=up^ν$. Let $σ_1,\dots,σ_n$ 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 $α\in L$ set $E_h(α) =e_h(σ_1(α),\dots,σ_n(α))$. Set $j=\min\{v_p(h),ν\}$. 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 $\mathcal{M}_K$.

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