{"paper":{"title":"Variations of the Primitive Normal Basis Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Giorgos Kapetanakis, Lucas Reis","submitted_at":"2017-12-28T13:58:59Z","abstract_excerpt":"The celebrated Primitive Normal Basis Theorem states that for any $n\\ge 2$ and any finite field $\\mathbb F_q$, there exists an element $\\alpha\\in \\mathbb F_{q^n}$ that is simultaneously primitive and normal over $\\mathbb F_q$. In this paper, we prove some variations of this result, completing the proof of a conjecture proposed by Anderson and Mullen (2014). Our results also imply the existence of elements of $\\mathbb F_{q^n}$ with multiplicative order $(q^n-1)/2$ and prescribed trace over $\\mathbb F_q$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.09861","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"}