{"title":"关于 k 法向元素数和 Fq 实用数","authors":"Josimar J.R. Aguirre, Victor G.L. Neumann","doi":"10.1016/j.ffa.2024.102444","DOIUrl":null,"url":null,"abstract":"<div><p>A normal element in a finite field extension <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> is characterized by having linearly independent conjugates over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. We consider the generalization of normal elements known as <em>k</em>-normal elements, where a subset of the conjugates are required to be linearly independent. In this paper, we provide an explicit combinatorial formula for counting the number of <em>k</em>-normal elements in a finite field extension motivated by an open problem proposed by Huczynska, Mullen, Panario, and Thomson in 2013. Furthermore, we use these results to establish new insights about <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-practical numbers.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the number of k-normal elements and Fq-practical numbers\",\"authors\":\"Josimar J.R. Aguirre, Victor G.L. Neumann\",\"doi\":\"10.1016/j.ffa.2024.102444\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A normal element in a finite field extension <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> is characterized by having linearly independent conjugates over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. We consider the generalization of normal elements known as <em>k</em>-normal elements, where a subset of the conjugates are required to be linearly independent. In this paper, we provide an explicit combinatorial formula for counting the number of <em>k</em>-normal elements in a finite field extension motivated by an open problem proposed by Huczynska, Mullen, Panario, and Thomson in 2013. Furthermore, we use these results to establish new insights about <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-practical numbers.</p></div>\",\"PeriodicalId\":50446,\"journal\":{\"name\":\"Finite Fields and Their Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-05-02\",\"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/S1071579724000832\",\"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/S1071579724000832","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
有限域扩展 Fqn/Fq 中的正则元的特征是在 Fq 上有线性独立的共轭。我们考虑正则元的广义化，即 k 正则元，其中要求共轭子集线性独立。在本文中，我们根据 Huczynska、Mullen、Panario 和 Thomson 于 2013 年提出的一个开放问题，提供了计算有限域扩展中 k 正则元素数量的明确组合公式。此外，我们还利用这些结果建立了关于 Fq 实用数的新见解。
On the number of k-normal elements and Fq-practical numbers
A normal element in a finite field extension is characterized by having linearly independent conjugates over . We consider the generalization of normal elements known as k-normal elements, where a subset of the conjugates are required to be linearly independent. In this paper, we provide an explicit combinatorial formula for counting the number of k-normal elements in a finite field extension motivated by an open problem proposed by Huczynska, Mullen, Panario, and Thomson in 2013. Furthermore, we use these results to establish new insights about -practical numbers.
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.
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