Translates of completely normal elements and the Morgan-Mullen conjecture
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917093504450560 |
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| author | Garefalakis, Theodoulos Kapetanakis, Giorgos |
| author_facet | Garefalakis, Theodoulos Kapetanakis, Giorgos |
| contents | 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 establish, under conditions, the existence of elements that are simultaneously primitive and $q^n/q$-completely normal, covering some yet unresolved cases of a 30-year-old conjecture by Morgan and Mullen. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23245 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Translates of completely normal elements and the Morgan-Mullen conjecture Garefalakis, Theodoulos Kapetanakis, Giorgos Number Theory 11T30, 11T23, 12E20 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 establish, under conditions, the existence of elements that are simultaneously primitive and $q^n/q$-completely normal, covering some yet unresolved cases of a 30-year-old conjecture by Morgan and Mullen. |
| title | Translates of completely normal elements and the Morgan-Mullen conjecture |
| topic | Number Theory 11T30, 11T23, 12E20 |
| url | https://arxiv.org/abs/2509.23245 |