{"paper":{"title":"Perturbing Eisenstein polynomials over local fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kevin Keating","submitted_at":"2017-01-08T16:18:26Z","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. Let $\\pi_L$ be a uniformizer for $L$ and let $f(X)$ be the minimum polynomial for $\\pi_L$ over $K$. Suppose $\\tilde{\\pi}_L$ is another uniformizer for $L$ such that $\\tilde{\\pi}_L\\equiv\\pi_L+r\\pi_L^{\\ell+1} \\pmod{\\pi_L^{\\ell+2}}$ for some $\\ell\\ge1$ and $r\\in O_K$. Let $\\tilde{f}(X)$ be the minimum polynomial for $\\tilde{\\pi}_L$ over $K$. In this paper we give congruences for the coefficients of $\\tilde{f}(X)$ in terms of $r$ and the coefficients of $f(X)$. These "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.01978","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"}