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，它提供了关于多项式通过根式可解性的信息。在过去的几年里，申请人在计算有理数上的中间域和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高两位数阶的多项式的伽罗瓦群的计算，但是这些计算难以有效地执行，并且不知道这些算法是否可以具有多项式时间复杂度。对于中间场的计算，申请人开发了一种新算法，该算法提供了以多项式复杂度生成子域的系统。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，计算所有子场的运行时间与子场的数量成比例。对于手头的这个项目，我们的目标是开发和实现非平凡的算法，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。这是公平的希望，这些算法提供了改进和更好地理解各自的算法在全球领域。这个项目需要一个非常详细的调查，当地的字段的结构，并检索有效性的计算在当地字段的一个良好的直觉。这产生了一个很好的理论和计算机代数应用的互动。计算中间领域导致回到因式分解的多项式和解决方案的线性方程组。因此，最近在局部域上实现的因式分解算法只提供近似值，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。局部域上的伽罗瓦群的计算的主要问题是我们不容易获得零点及其近似，这不允许对全局域的已知算法进行变换。我们希望通过绝对伽罗瓦群和局部类域理论的知识来攻击伽罗瓦群的计算。申请人管理一个数据库，用于填写超过200万个多项式的数字字段。该数据库应通过局部功能字段进行扩展，并具有较小的特征。这些数据对于发现和理解几何和数论中的几何图形的有趣例子非常重要。此外，大表格对于制作和检验关于此类对象的渐近性的图表是有用的。