Computational Galois Theory for Local Fields

局部域的计算伽罗瓦理论

• 批准号: 239392052
• 负责人: Professor Dr. Jürgen Klüners
• 金额: --
• 依托单位: Fachgebiet Computeralgebra und Zahlentheorie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2013
• 资助国家: 德国
• 起止时间: 2012-12-31 至 2016-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/239392052?language=en
• 关键词: Computational; Galois; Theory; Local; Fields

Galois groups are fundamental mathematical objects, which provide information about the solvability of polynomials by radicals. The applicant has gained respectable progress in computing intermediate fields and Galois groups over rational numbers in the past few years. While the recent implementations provides computations of Galois groups for polynomials up to high double-digit degree, these computations are difficult to perform efficiently, and it is unknown if these algorithms can have polynomial time complexity. For the computation of intermediate fields, the applicant developed a new algorithm, which de-livers a system of generating subfields in polynomial complexity. Then any arbitrary subfield can be described as a suitable intersection of the generating subfields. Furthermore, the running time for computing all subfields is proportional to the number of subfields. For this project at hand, we aim to develop and implement nontrivial algorithms for the computation of subfields and Galois groups over local fields, i.e. over p-adic fields and local function fields. It is fair to hope that these algorithms provide improvements and better understanding for the respective algorithms over global fields. This project requires a very detailed investigation of the local fields' structure, and a fine intuition for retrieving effectiveness for computation over local fields. This yields a nice interaction of theory and computer algebra applications. The computation of intermediate fields leads back to factorisation of polynomials and solutions of linear systems of equations. Hereby, recent implementations of factorisation algorithms over local fields only provide approximations, and as these are the input of the linear system of equations, we have to consider precision problems like in numerical analysis. The main problem for the computation of Galois groups over local fields is that we do not have easy access to the zeros and their approximations, which does not allow the transformation of known algorithms for global fields. We would like to attack the computation of Galois group via the knowledge of the absolute Galois group and local class field theory. The applicant administrates a database for number fields filled with over 2 million polynomials. This database shall be extended and be expanded by local function fields with small characteristic. These data are very important to find and understand interesting examples for conjectures within geometry and number theory. Furthermore, big tables are useful to make and test conjectures about the asymptotics of such objects.

伽罗瓦群是基本的数学对象，它提供有关根式多项式可解性的信息。申请人在过去几年中在计算中间域和有理数上的伽罗瓦群方面取得了可观的进展。虽然最近的实现提供了高达两位数次数的多项式的伽罗瓦群计算，但这些计算很难有效地执行，并且未知这些算法是否可以具有多项式时间复杂度。对于中间字段的计算，申请人开发了一种新算法，其提供了以多项式复杂度生成子字段的系统。然后任何任意子场都可以被描述为生成子场的适当交集。此外，计算所有子字段的运行时间与子字段的数量成正比。对于手头的这个项目，我们的目标是开发和实现非平凡算法，用于计算局部域上的子域和伽罗瓦群，即 p-adic 域和局部函数域。公平地希望这些算法能够为全球领域的各个算法提供改进和更好的理解。该项目需要对局部域的结构进行非常详细的研究，以及检索局部域计算有效性的良好直觉。这产生了理论和计算机代数应用的良好交互。中间场的计算导致多项式的因式分解和线性方程组的解。因此，最近在局部域上实现的因式分解算法仅提供近似值，并且由于这些是线性方程组的输入，因此我们必须考虑数值分析中的精度问题。局部域上伽罗瓦群计算的主要问题是我们无法轻松获得零点及其近似值，这不允许将已知算法转换为全局域。我们想通过绝对伽罗瓦群和局部类域论的知识来攻击伽罗瓦群的计算。申请人管理一个包含超过 200 万个多项式的数字字段的数据库。该数据库应通过局部特征较小的功能域进行扩展和扩展。这些数据对于查找和理解几何和数论中猜想的有趣示例非常重要。此外，大表对于做出和测试有关此类对象的渐近性的猜想很有用。