Elements of high order in finite fields specified by binomials
Let $F_q$ be a field with $q$ elements, where $q$ is a power of a prime number $p\geq 5$. For any integer $m\geq 2$ and $a\in F_q^*$ such that the polynomial $x^m-a$ is irreducible in $F_q[x]$, we combine two different methods to explicitly construct elements of high order in the field $F_q[x]/\lang...
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| Format: | Article |
| Language: | English |
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Vasyl Stefanyk Precarpathian National University
2022-06-01
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| Series: | Karpatsʹkì Matematičnì Publìkacìï |
| Subjects: | |
| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/5880 |
| _version_ | 1797205796329619456 |
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| author | V. Bovdi A. Diene R. Popovych |
| author_facet | V. Bovdi A. Diene R. Popovych |
| author_sort | V. Bovdi |
| collection | DOAJ |
| description | Let $F_q$ be a field with $q$ elements, where $q$ is a power of a prime number $p\geq 5$. For any integer $m\geq 2$ and $a\in F_q^*$ such that the polynomial $x^m-a$ is irreducible in $F_q[x]$, we combine two different methods to explicitly construct elements of high order in the field $F_q[x]/\langle x^m-a\rangle$. Namely, we find elements with multiplicative order of at least $5^{\sqrt[3]{m/2}}$, which is better than previously obtained bound for such family of extension fields. |
| first_indexed | 2024-04-24T08:56:49Z |
| format | Article |
| id | doaj.art-8af9f72ffbbf4022ba42559e4b10c016 |
| institution | Directory Open Access Journal |
| issn | 2075-9827 2313-0210 |
| language | English |
| last_indexed | 2024-04-24T08:56:49Z |
| publishDate | 2022-06-01 |
| publisher | Vasyl Stefanyk Precarpathian National University |
| record_format | Article |
| series | Karpatsʹkì Matematičnì Publìkacìï |
| spelling | doaj.art-8af9f72ffbbf4022ba42559e4b10c0162024-04-16T07:10:59ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102022-06-0114123824610.15330/cmp.14.1.238-2465085Elements of high order in finite fields specified by binomialsV. Bovdi0https://orcid.org/0000-0001-5750-163XA. Diene1R. Popovych2United Arab Emirates University, Al Ain, United Arab EmiratesUnited Arab Emirates University, Al Ain, United Arab EmiratesLviv Polytechnic National University, 12 Bandera str., 79013, Lviv, UkraineLet $F_q$ be a field with $q$ elements, where $q$ is a power of a prime number $p\geq 5$. For any integer $m\geq 2$ and $a\in F_q^*$ such that the polynomial $x^m-a$ is irreducible in $F_q[x]$, we combine two different methods to explicitly construct elements of high order in the field $F_q[x]/\langle x^m-a\rangle$. Namely, we find elements with multiplicative order of at least $5^{\sqrt[3]{m/2}}$, which is better than previously obtained bound for such family of extension fields.https://journals.pnu.edu.ua/index.php/cmp/article/view/5880finite fieldmultiplicative orderelement of high multiplicative orderbinomial |
| spellingShingle | V. Bovdi A. Diene R. Popovych Elements of high order in finite fields specified by binomials Karpatsʹkì Matematičnì Publìkacìï finite field multiplicative order element of high multiplicative order binomial |
| title | Elements of high order in finite fields specified by binomials |
| title_full | Elements of high order in finite fields specified by binomials |
| title_fullStr | Elements of high order in finite fields specified by binomials |
| title_full_unstemmed | Elements of high order in finite fields specified by binomials |
| title_short | Elements of high order in finite fields specified by binomials |
| title_sort | elements of high order in finite fields specified by binomials |
| topic | finite field multiplicative order element of high multiplicative order binomial |
| url | https://journals.pnu.edu.ua/index.php/cmp/article/view/5880 |
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