On index divisors and monogenity of certain octic number fields defined by $x^8+ax^3+b$
UDC 511 For any octic number field $K$ generated by a root $\alpha$ of a monic irreducible trinomial $F(x)=x^8+ax^3+b \in \mathbb{Z}[x]$ and for every rational prime $p,$ we show when $p$ divides the index of $K.$  We also describe the prime power decomposition of the index...
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| Datum: | 2024 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Institute of Mathematics, NAS of Ukraine
2024
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/7536 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512681998417920 |
|---|---|
| author | Kchit, Omar Kchit, Omar |
| author_facet | Kchit, Omar Kchit, Omar |
| author_sort | Kchit, Omar |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-08-10T06:39:16Z |
| description | UDC 511
For any octic number field $K$ generated by a root $\alpha$ of a monic irreducible trinomial $F(x)=x^8+ax^3+b \in \mathbb{Z}[x]$ and for every rational prime $p,$ we show when $p$ divides the index of $K.$  We also describe the prime power decomposition of the index $i(K).$ In this way, we give a partial answer to { Problem $22$} of Narkiewicz [Elementary and analytic theory of algebraic numbers, Springer-Verlag, Auflage (2004)] for this family of number fields. As an application of our results, we conclude that if $i(K)\neq1,$  then $K$ is not monogenic. We illustrate our results by some computational examples.  |
| doi_str_mv | 10.3842/umzh.v76i7.7536 |
| first_indexed | 2026-03-24T03:32:40Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-7536 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:32:40Z |
| publishDate | 2024 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-75362024-08-10T06:39:16Z On index divisors and monogenity of certain octic number fields defined by $x^8+ax^3+b$ On index divisors and monogenity of certain octic number fields defined by $x^8+ax^3+b$ Kchit, Omar Kchit, Omar Theorem of Dedekind Theorem of Ore prime ideal factorization Newton polygon Index of a number field Power integral basis Monogenic UDC 511 For any octic number field $K$ generated by a root $\alpha$ of a monic irreducible trinomial $F(x)=x^8+ax^3+b \in \mathbb{Z}[x]$ and for every rational prime $p,$ we show when $p$ divides the index of $K.$  We also describe the prime power decomposition of the index $i(K).$ In this way, we give a partial answer to { Problem $22$} of Narkiewicz [Elementary and analytic theory of algebraic numbers, Springer-Verlag, Auflage (2004)] for this family of number fields. As an application of our results, we conclude that if $i(K)\neq1,$  then $K$ is not monogenic. We illustrate our results by some computational examples.  УДК 511 Про індексні дільники та моногенність деяких полів октичних чисел, що задані формулою $x^8+ax^3+b$  Для довільного октичного числового поля $K$, породженого коренем $\alpha$ монічного незвідного тричлена $F(x)=x^8+ax^3+b \in \mathbb{Z}[x],$ і для кожного раціонального простого $p$ показано, коли $p$ ділить індекс $K$. Також описано розклад індексу $i(K)$ за простими степенями. Таким чином, дано часткову відповідь на {проблему $22$} Наркевича [Elementary and analytic theory of algebraic numbers, Springer-Verlag, Auflage (2004)] для цієї сім'ї числових полів. Як застосування одержаних результатів, зроблено висновок, що якщо $i(K)\neq1,$ то $K$ не є моногенним. Отримані результати проілюстровано деякими обчислювальними прикладами. Institute of Mathematics, NAS of Ukraine 2024-08-04 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7536 10.3842/umzh.v76i7.7536 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 7 (2024); 992 - 1006 Український математичний журнал; Том 76 № 7 (2024); 992 - 1006 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7536/10055 Copyright (c) 2024 Omar Kchit |
| spellingShingle | Kchit, Omar Kchit, Omar On index divisors and monogenity of certain octic number fields defined by $x^8+ax^3+b$ |
| title | On index divisors and monogenity of certain octic number fields defined by $x^8+ax^3+b$ |
| title_alt | On index divisors and monogenity of certain octic number fields defined by $x^8+ax^3+b$ |
| title_full | On index divisors and monogenity of certain octic number fields defined by $x^8+ax^3+b$ |
| title_fullStr | On index divisors and monogenity of certain octic number fields defined by $x^8+ax^3+b$ |
| title_full_unstemmed | On index divisors and monogenity of certain octic number fields defined by $x^8+ax^3+b$ |
| title_short | On index divisors and monogenity of certain octic number fields defined by $x^8+ax^3+b$ |
| title_sort | on index divisors and monogenity of certain octic number fields defined by $x^8+ax^3+b$ |
| topic_facet | Theorem of Dedekind Theorem of Ore prime ideal factorization Newton polygon Index of a number field Power integral basis Monogenic |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7536 |
| work_keys_str_mv | AT kchitomar onindexdivisorsandmonogenityofcertainocticnumberfieldsdefinedbyx8ax3b AT kchitomar onindexdivisorsandmonogenityofcertainocticnumberfieldsdefinedbyx8ax3b |