{"paper":{"title":"Extensions of local fields and elementary symmetric polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kevin Keating","submitted_at":"2016-08-26T02:13:58Z","abstract_excerpt":"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 $E_h(\\mathcal{M}_L^r)\\subset \\mathcal{M}_K^{\\lceil(i_j+hr)/n\\rceil}$, wh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.07350","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}