{"paper":{"title":"Existence of primitive $1$-normal elements in finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"David Thomson, Lucas Reis","submitted_at":"2017-10-17T07:21:33Z","abstract_excerpt":"An element $\\alpha \\in \\mathbb F_{q^n}$ is \\emph{normal} if $\\mathcal{B} = \\{\\alpha, \\alpha^q, \\ldots, \\alpha^{q^{n-1}}\\}$ forms a basis of $\\mathbb F_{q^n}$ as a vector space over $\\mathbb F_{q}$; in this case, $\\mathcal{B}$ is a normal basis of $\\mathbb F_{q^n}$ over $\\mathbb F_{q}$. The notion of $k$-normal elements was introduced in Huczynska et al (2013). Using the same notation as before, $\\alpha$ is $k$-normal if $\\mathcal{B}$ spans a co-dimension $k$ subspace of $\\mathbb F_{q^n}$. It can be shown that $1$-normal elements always exist in $\\mathbb F_{q^n}$, and Huczynska et al (2013) sho"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.06131","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"}