{"paper":{"title":"Nilpotent and abelian Hopf-Galois structures on field extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Nigel P. Byott","submitted_at":"2012-10-05T09:58:03Z","abstract_excerpt":"Let $L/K$ be a finite Galois extension of fields with group $\\Gamma$. When $\\Gamma$ is nilpotent, we show that the problem of enumerating all nilpotent Hopf-Galois structures on $L/K$ can be reduced to the corresponding problem for the Sylow subgroups of $\\Gamma$. We use this to enumerate all nilpotent (resp. abelian) Hopf-Galois structures on a cyclic extension of arbitrary finite degree. When $\\Gamma$ is abelian, we give conditions under which every abelian Hopf-Galois structure on $L/K$ has type $\\Gamma$. We also give a criterion on $n$ such that \\emph{every} Hopf-Galois structure on a cycl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.1693","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"}