{"paper":{"title":"Integral bases and monogenity of the simplest sextic fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Istv\\'an Ga\\'al, L\\'aszl\\'o Remete","submitted_at":"2018-09-26T15:46:02Z","abstract_excerpt":"Let $m$ be an integer, $m\\neq -8,-3,0,5$ such that $m^2+3m+9$ is square free. Let $\\alpha$ be a root of \\[ f=x^6-2mx^5-(5m+15)x^4-20x^3+5mx^2+(2m+6)x+1. \\] The totally real cyclic fields $K=Q(\\alpha)$ are called simplest sextic fields and are well known in the literature.\n  Using a completely new approach we explicitly give an integral basis of $K$ in a parametric form and we show that the structure of this integral basis is periodic in $m$ with period length 36. We prove that $K$ is not monogenic except for a few values of $m$ in which cases we give all generators of power integral bases."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.10072","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"}