{"paper":{"title":"Translates of completely normal elements and the Morgan-Mullen conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Giorgos Kapetanakis, Theodoulos Garefalakis","submitted_at":"2025-09-27T10:51:02Z","abstract_excerpt":"Denote by $\\mathbb F_q$ the finite field of order $q$ and by $\\mathbb F_{q^n}$ its extension of degree $n$. Some $a\\in\\mathbb F_{q^n}$ is called primitive if it generates the multiplicative group $\\mathbb F_{q^n}^*$ and it is called $q^n/q$-normal if its $\\mathbb F_q$-conjugates form an $\\mathbb F_q$-basis of $\\mathbb F_{q^n}$ if the latter is viewed as an $\\mathbb F_q$-vector space. Furthermore, some $a\\in\\mathbb F_{q^n}$ is called $q^n/q$-completely normal if it is $q^n/q^d$-normal for all $d\\mid n$. In this work we prove a new construction of sets of completely normal elements and, we estab"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.23245","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.23245/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}