{"paper":{"title":"Primitive Element Pairs with One Prescribed Trace over a Finite Field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.NT","authors_text":"Anju Gupta, R. K. Sharma, Stephen D. Cohen","submitted_at":"2017-09-16T16:58:16Z","abstract_excerpt":"In this article, we establish a sufficient condition for the existence of a primitive element $\\alpha \\in {\\mathbb{F}_{q^n}}$ such that the element $\\alpha+\\alpha^{-1}$ is also a primitive element of ${\\mathbb{F}_{q^n}},$ and $Tr_{\\mathbb{F}_{q^n}|\\mathbb{F}_{q}}(\\alpha)=a$ for any prescribed $a \\in \\mathbb{F}_q$, where $q=p^k$ for some prime $p$ and positive integer $k$. We prove that every finite field $\\mathbb{F}_{q^n}~ (n \\geq5),$ contains such primitive elements except for finitely many values of $q$ and $n$. Indeed, by computation, we conclude that there are no actual exceptional pairs $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.05540","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"}