幂组合多项式的指数

IF 1.2 3区 数学 Q1 MATHEMATICS
Sumandeep Kaur , Surender Kumar , László Remete
{"title":"幂组合多项式的指数","authors":"Sumandeep Kaur ,&nbsp;Surender Kumar ,&nbsp;László Remete","doi":"10.1016/j.ffa.2025.102642","DOIUrl":null,"url":null,"abstract":"<div><div>The index of a monic irreducible polynomial <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> having a root <em>θ</em> is the index <span><math><mo>[</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>:</mo><mi>Z</mi><mo>[</mo><mi>θ</mi><mo>]</mo><mo>]</mo></math></span> where <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> is the ring of algebraic integers of the number field <span><math><mi>K</mi><mo>=</mo><mi>Q</mi><mo>(</mo><mi>θ</mi><mo>)</mo></math></span>. If <span><math><mo>[</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>:</mo><mi>Z</mi><mo>[</mo><mi>θ</mi><mo>]</mo><mo>]</mo><mo>=</mo><mn>1</mn></math></span>, then <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is monogenic. In this paper, we give necessary and sufficient conditions for a monic irreducible power compositional polynomial <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span> belonging to <span><math><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, to be monogenic. As an application of our results, for a polynomial <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>+</mo><mi>A</mi><mo>⋅</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, with <span><math><mi>d</mi><mo>&gt;</mo><mn>1</mn><mo>,</mo><mi>deg</mi><mo>⁡</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>&lt;</mo><mi>d</mi></math></span> and <span><math><mo>|</mo><mi>h</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>|</mo><mo>=</mo><mn>1</mn></math></span>, we prove that for each positive integer <em>k</em> with <span><math><mi>rad</mi><mo>(</mo><mi>k</mi><mo>)</mo><mo>|</mo><mi>rad</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, the power compositional polynomial <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span> is monogenic if and only if <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is monogenic, provided that <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span> is irreducible. At the end of the paper, we give infinite families of polynomials as examples.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102642"},"PeriodicalIF":1.2000,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the index of power compositional polynomials\",\"authors\":\"Sumandeep Kaur ,&nbsp;Surender Kumar ,&nbsp;László Remete\",\"doi\":\"10.1016/j.ffa.2025.102642\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>The index of a monic irreducible polynomial <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> having a root <em>θ</em> is the index <span><math><mo>[</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>:</mo><mi>Z</mi><mo>[</mo><mi>θ</mi><mo>]</mo><mo>]</mo></math></span> where <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> is the ring of algebraic integers of the number field <span><math><mi>K</mi><mo>=</mo><mi>Q</mi><mo>(</mo><mi>θ</mi><mo>)</mo></math></span>. If <span><math><mo>[</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>:</mo><mi>Z</mi><mo>[</mo><mi>θ</mi><mo>]</mo><mo>]</mo><mo>=</mo><mn>1</mn></math></span>, then <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is monogenic. In this paper, we give necessary and sufficient conditions for a monic irreducible power compositional polynomial <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span> belonging to <span><math><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, to be monogenic. As an application of our results, for a polynomial <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>+</mo><mi>A</mi><mo>⋅</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, with <span><math><mi>d</mi><mo>&gt;</mo><mn>1</mn><mo>,</mo><mi>deg</mi><mo>⁡</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>&lt;</mo><mi>d</mi></math></span> and <span><math><mo>|</mo><mi>h</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>|</mo><mo>=</mo><mn>1</mn></math></span>, we prove that for each positive integer <em>k</em> with <span><math><mi>rad</mi><mo>(</mo><mi>k</mi><mo>)</mo><mo>|</mo><mi>rad</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, the power compositional polynomial <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span> is monogenic if and only if <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is monogenic, provided that <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span> is irreducible. At the end of the paper, we give infinite families of polynomials as examples.</div></div>\",\"PeriodicalId\":50446,\"journal\":{\"name\":\"Finite Fields and Their Applications\",\"volume\":\"107 \",\"pages\":\"Article 102642\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2025-05-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Finite Fields and Their Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1071579725000723\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Finite Fields and Their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1071579725000723","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

具有根θ的一元不可约多项式f(x)∈Z[x]的索引是索引[ZK:Z[θ]],其中ZK是数域K=Q(θ)的代数整数环。如果[ZK:Z[θ]]=1,则f(x)是单基因的。本文给出了属于Z[x]的一元不可约幂组合多项式f(xk)是单原的充要条件。作为结果的一个应用,对于多项式f(x)=xd+ a·h(x)∈Z[x],当d>;1,deg (h(x)<d, |h(0)|=1时,证明了对于每一个正整数k,当且仅当f(x)是单基因的,当f(xk)不可约时,幂复合多项式f(xk)是单基因的。在文章的最后,我们给出了无限多项式族作为例子。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On the index of power compositional polynomials
The index of a monic irreducible polynomial f(x)Z[x] having a root θ is the index [ZK:Z[θ]] where ZK is the ring of algebraic integers of the number field K=Q(θ). If [ZK:Z[θ]]=1, then f(x) is monogenic. In this paper, we give necessary and sufficient conditions for a monic irreducible power compositional polynomial f(xk) belonging to Z[x], to be monogenic. As an application of our results, for a polynomial f(x)=xd+Ah(x)Z[x], with d>1,degh(x)<d and |h(0)|=1, we prove that for each positive integer k with rad(k)|rad(A), the power compositional polynomial f(xk) is monogenic if and only if f(x) is monogenic, provided that f(xk) is irreducible. At the end of the paper, we give infinite families of polynomials as examples.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
2.00
自引率
20.00%
发文量
133
审稿时长
6-12 weeks
期刊介绍: Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering. For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods. The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信