{"paper":{"title":"Lower Bounds for Heights in Relative Galois Extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alia Hamieh, Kathleen Petersen, Kevser Akta\\c{s}, Kirsti Biggs, Lola Thompson, Shabnam Akhtari","submitted_at":"2017-04-10T18:03:41Z","abstract_excerpt":"The goal of this paper is to obtain lower bounds on the height of an algebraic number in a relative setting, extending previous work of Amoroso and Masser. Specifically, in our first theorem we obtain an effective bound for the height of an algebraic number $\\alpha$ when the base field $\\mathbb{K}$ is a number field and $\\mathbb{K}(\\alpha)/\\mathbb{K}$ is Galois. Our second result establishes an explicit height bound for any non-zero element $\\alpha$ which is not a root of unity in a Galois extension $\\mathbb{F}/\\mathbb{K}$, depending on the degree of $\\mathbb{K}/\\mathbb{Q}$ and the number of c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.02995","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"}