{"paper":{"title":"A family of pairs of imaginary cyclic fields of degree $(p-1)/2$ with both class numbers divisible by $p$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Miho Aoki, Yasuhiro Kishi","submitted_at":"2018-09-21T08:25:27Z","abstract_excerpt":"We construct a new infinite family of pairs of imaginary cyclic fields of degree $(p-1)/2$ explicitly with both class numbers divisible by a given prime number $p$. For the proof, we use the fundamental unit of $\\mathbb Q(\\sqrt{p})$, certain units which are roots of a parametric quartic polynomial, the Kummer theory, the Gauss sums and the Jacobi sums, linear recurrence sequences, a consequence of the Weil conjecture and a result of Lenstra which is a generalization of Artin conjecture on primitive roots. Our result is based on the famous Scholz' results on pairs of quadratic fields $\\mathbb Q"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.07982","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"}