Units in the group algebra $FS_{3}$
UDC 512.552 We explicitly describe each unit of a group algebra $Z_{p} S_{3}$ for each positive prime $p \geq 5$ by using a characterization of the group algebra of the metacyclic group $G= \langle x,c\colon x^{3}=1,\ c^{n}=1,\ cxc^{-1 } = x^{-1} \rangle$ over the finite field $F$ of characteristic...
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| Datum: | 2026 |
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| Sprache: | Englisch |
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Institute of Mathematics, NAS of Ukraine
2026
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/9213 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860513449942974464 |
|---|---|
| author | Gupta, Abhinay Kumar Sharma, R. K. Gupta, Abhinay Kumar Sharma, R. K. |
| author_facet | Gupta, Abhinay Kumar Sharma, R. K. Gupta, Abhinay Kumar Sharma, R. K. |
| author_sort | Gupta, Abhinay Kumar |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-03-21T11:04:14Z |
| description | UDC 512.552
We explicitly describe each unit of a group algebra $Z_{p} S_{3}$ for each positive prime $p \geq 5$ by using a characterization of the group algebra of the metacyclic group $G= \langle x,c\colon x^{3}=1,\ c^{n}=1,\ cxc^{-1 } = x^{-1} \rangle$ over the finite field $F$ of characteristic $p,$ where $p$ is a positive prime such that $p \nmid 3n.$ Based on our findings, we pose a conjecture on the number of roots of some explicit polynomials over the prime field $\mathbb{Z}_{p}$ for further academic explorations. |
| doi_str_mv | 10.3842/umzh.v78i1-2.9213 |
| first_indexed | 2026-03-24T03:44:52Z |
| format | Article |
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| id | umjimathkievua-article-9213 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:44:52Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-92132026-03-21T11:04:14Z Units in the group algebra $FS_{3}$ Units in the group algebra $FS_{3}$ Gupta, Abhinay Kumar Sharma, R. K. Gupta, Abhinay Kumar Sharma, R. K. Group rings; Unit group; Prime fields; Polynomials. Primary 16U60; Secondary 20C05, 16S34 UDC 512.552 We explicitly describe each unit of a group algebra $Z_{p} S_{3}$ for each positive prime $p \geq 5$ by using a characterization of the group algebra of the metacyclic group $G= \langle x,c\colon x^{3}=1,\ c^{n}=1,\ cxc^{-1 } = x^{-1} \rangle$ over the finite field $F$ of characteristic $p,$ where $p$ is a positive prime such that $p \nmid 3n.$ Based on our findings, we pose a conjecture on the number of roots of some explicit polynomials over the prime field $\mathbb{Z}_{p}$ for further academic explorations. УДК 512.552 Оборотні елементи у груповій алгебрі $F S_{3}$ Явно описано кожний оборотний елемент групової алгебри $Z_{p} S_{3}$ для кожного простого числа $p \geq 5$ з використанням характеристики групової алгебри метациклічної групи $G = \langle x,c\colon x^{3}=1,\ c^{n}=1,\ cxc^{-1 } = x^{-1} \rangle$ над скінченним полем $F$ з характеристикою $p,$ де $p$ – додатне просте число таке, що $p \nmid 3n.$ На підставі отриманих результатів висунуто гіпотезу щодо кількості коренів деяких явних многочленів над простим полем $\mathbb{Z}_{p}$ для подальшого наукового опрацювання. Institute of Mathematics, NAS of Ukraine 2026-03-02 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/9213 10.3842/umzh.v78i1-2.9213 Ukrains’kyi Matematychnyi Zhurnal; Vol. 78 No. 1-2 (2026); 83 Український математичний журнал; Том 78 № 1-2 (2026); 83 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/9213/10623 Copyright (c) 2026 Abhinay Kumar Gupta, R. K. Sharma |
| spellingShingle | Gupta, Abhinay Kumar Sharma, R. K. Gupta, Abhinay Kumar Sharma, R. K. Units in the group algebra $FS_{3}$ |
| title | Units in the group algebra $FS_{3}$ |
| title_alt | Units in the group algebra $FS_{3}$ |
| title_full | Units in the group algebra $FS_{3}$ |
| title_fullStr | Units in the group algebra $FS_{3}$ |
| title_full_unstemmed | Units in the group algebra $FS_{3}$ |
| title_short | Units in the group algebra $FS_{3}$ |
| title_sort | units in the group algebra $fs_{3}$ |
| topic_facet | Group rings Unit group Prime fields Polynomials. Primary 16U60; Secondary 20C05 16S34 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/9213 |
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