{"paper":{"title":"Further results on 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":"2018-11-01T10:51:33Z","abstract_excerpt":"Let $\\mathbb{F}_q$ be the finite field of characteristic $p$ with $q$ elements and $\\mathbb{F}_{q^n}$ its extension of degree $n$. The conjecture of Morgan and Mullen asserts the existence of primitive and completely normal elements (PCN elements) for the extension $\\mathbb{F}_{q^n}/\\mathbb{F}_q$ for any $q$ and $n$. It is known that the conjecture holds for $n \\leq q$. In this work we prove the conjecture for a larger range of exponents. In particular, we give sharper bounds for the number of completely normal elements and use them to prove asymptotic and effective existence results for $q\\le"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.00896","kind":"arxiv","version":4},"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"}