{"paper":{"title":"Gr\\\"obner Bases over Algebraic Number Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SC","math.NT"],"primary_cat":"math.AC","authors_text":"Andreas Steenpass, Claus Fieker, Dereje Kifle Boku, Wolfram Decker","submitted_at":"2015-04-17T16:44:30Z","abstract_excerpt":"Although Buchberger's algorithm, in theory, allows us to compute Gr\\\"obner bases over any field, in practice, however, the computational efficiency depends on the arithmetic of the ground field. Consider a field $K = \\mathbb{Q}(\\alpha)$, a simple extension of $\\mathbb{Q}$, where $\\alpha$ is an algebraic number, and let $f \\in \\mathbb{Q}[t]$ be the minimal polynomial of $\\alpha$. In this paper we present a new efficient method to compute Gr\\\"obner bases in polynomial rings over the algebraic number field $K$. Starting from the ideas of Noro [Noro, 2006], we proceed by joining $f$ to the ideal t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04564","kind":"arxiv","version":2},"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"}