# Kevin Keating

Affiliation: University of Florida

Email: keating@ufl.edu

Title Of Talk: Extensions of local fields and elementary symmetric polynomials

Abstract: Let $K$ be a local field with valuation $v_K$. Let $K^{sep}$ be a separable closure of $K$, and let $L/K$ be a finite totally ramified subextension of $K^{sep}/K$ of degree $n$. Let $\sigma_1,\dots,\sigma_n$ denote the $K$-embeddings of $L$ into $K^{sep}$. For $1\le i\le n$ let $s_i(X_1,\dots,X_n)$ denote the $i$th elementary symmetric polynomial in $n$ variables, and for $\alpha\in L$ set $S_i(\alpha)=s_i(\sigma_1(\alpha),\dots,\sigma_n(\alpha))$. In this talk we consider the problem of finding a lower bound for $v_K(S_i(\alpha))$ in terms of $v_L(\alpha)$. The solution seems to depend on the indices of inseparability of $L/K$.