Darien Connolly , Calvin George , Xiang-dong Hou , Adam Madro , Vincenzo Pallozzi Lavorante
{"title":"通过对称化和对称还原实现正多项式的方法","authors":"Darien Connolly , Calvin George , Xiang-dong Hou , Adam Madro , Vincenzo Pallozzi Lavorante","doi":"10.1016/j.ffa.2024.102525","DOIUrl":null,"url":null,"abstract":"<div><div>An irreducible polynomial <span><math><mi>f</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>X</mi><mo>]</mo></math></span> of degree <em>n</em> is <em>normal</em> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> if and only if its roots <span><math><mi>r</mi><mo>,</mo><msup><mrow><mi>r</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>r</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></msup></math></span> satisfy the condition <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>,</mo><msup><mrow><mi>r</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>r</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></msup><mo>)</mo><mo>≠</mo><mn>0</mn></math></span>, where <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span> is the <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> circulant determinant. By finding a suitable <em>symmetrization</em> of <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> (A multiple of <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> which is symmetric in <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>), we obtain a condition on the coefficients of <em>f</em> that is sufficient for <em>f</em> to be normal. This approach works well for <span><math><mi>n</mi><mo>≤</mo><mn>5</mn></math></span> but encounters computational difficulties when <span><math><mi>n</mi><mo>≥</mo><mn>6</mn></math></span>. In the present paper, we consider irreducible polynomials of the form <span><math><mi>f</mi><mo>=</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mi>a</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>X</mi><mo>]</mo></math></span>. For <span><math><mi>n</mi><mo>=</mo><mn>6</mn></math></span> and 7, by an indirect method, we are able to find simple conditions on <em>a</em> that are sufficient for <em>f</em> to be normal. In a more general context, we also explore the normal polynomials of a finite Galois extension through the irreducible characters of the Galois group.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102525"},"PeriodicalIF":1.2000,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"An approach to normal polynomials through symmetrization and symmetric reduction\",\"authors\":\"Darien Connolly , Calvin George , Xiang-dong Hou , Adam Madro , Vincenzo Pallozzi Lavorante\",\"doi\":\"10.1016/j.ffa.2024.102525\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>An irreducible polynomial <span><math><mi>f</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>X</mi><mo>]</mo></math></span> of degree <em>n</em> is <em>normal</em> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> if and only if its roots <span><math><mi>r</mi><mo>,</mo><msup><mrow><mi>r</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>r</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></msup></math></span> satisfy the condition <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>,</mo><msup><mrow><mi>r</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>r</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></msup><mo>)</mo><mo>≠</mo><mn>0</mn></math></span>, where <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span> is the <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> circulant determinant. By finding a suitable <em>symmetrization</em> of <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> (A multiple of <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> which is symmetric in <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>), we obtain a condition on the coefficients of <em>f</em> that is sufficient for <em>f</em> to be normal. This approach works well for <span><math><mi>n</mi><mo>≤</mo><mn>5</mn></math></span> but encounters computational difficulties when <span><math><mi>n</mi><mo>≥</mo><mn>6</mn></math></span>. In the present paper, we consider irreducible polynomials of the form <span><math><mi>f</mi><mo>=</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mi>a</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>X</mi><mo>]</mo></math></span>. For <span><math><mi>n</mi><mo>=</mo><mn>6</mn></math></span> and 7, by an indirect method, we are able to find simple conditions on <em>a</em> that are sufficient for <em>f</em> to be normal. In a more general context, we also explore the normal polynomials of a finite Galois extension through the irreducible characters of the Galois group.</div></div>\",\"PeriodicalId\":50446,\"journal\":{\"name\":\"Finite Fields and Their Applications\",\"volume\":\"101 \",\"pages\":\"Article 102525\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-10-23\",\"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/S1071579724001643\",\"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/S1071579724001643","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
当且仅当一个阶数为 n 的不可减多项式 f∈Fq[X] 的根 r,rq,...,rqn-1满足条件 Δn(r,rq,...,rqn-1)≠0,其中 Δn(X0,...,Xn-1)是 n×n 循环行列式时,这个 f∈Fq[X] 在 Fq 上是正常的。通过找到 Δn 的合适对称性(在 X0,...,Xn-1 中对称的 Δn 的倍数),我们就能得到 f 的系数条件,该条件足以保证 f 是正态的。这种方法在 n≤5 时效果很好,但在 n≥6 时遇到了计算上的困难。在本文中,我们考虑 f=Xn+Xn-1+a∈Fq[X] 形式的不可约多项式。对于 n=6 和 7,通过间接方法,我们能够找到关于 a 的简单条件,这些条件足以使 f 成为正多边形。在更一般的情况下,我们还通过伽罗瓦群的不可还原字符来探索有限伽罗瓦扩展的正多项式。
An approach to normal polynomials through symmetrization and symmetric reduction
An irreducible polynomial of degree n is normal over if and only if its roots satisfy the condition , where is the circulant determinant. By finding a suitable symmetrization of (A multiple of which is symmetric in ), we obtain a condition on the coefficients of f that is sufficient for f to be normal. This approach works well for but encounters computational difficulties when . In the present paper, we consider irreducible polynomials of the form . For and 7, by an indirect method, we are able to find simple conditions on a that are sufficient for f to be normal. In a more general context, we also explore the normal polynomials of a finite Galois extension through the irreducible characters of the Galois group.
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.