Computational Galois Theory for Local Fields

伽罗华群是基本的数学对象，它提供了关于根的多项式的可解性的信息。在过去的几年里，申请者在计算中间场和伽罗华群对有理数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的多项式的Galois群的计算，但这些计算很难有效地执行，并且这些算法是否具有多项式的时间复杂性是未知的。对于中间场的计算，申请人开发了一种新的算法，它描述了一个以多项式复杂性生成子场的系统。则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正比。在这个项目中，我们的目标是开发和实现计算局部域上的子域和伽罗华群的非平凡算法，即局部域上的子域和伽罗华群。希望这些算法能为全球领域的各个算法提供改进和更好的理解，这是公平的。这个项目需要对局域场的结构进行非常详细的调查，并对检索局域场计算的有效性有很好的直觉。这产生了理论和计算机代数应用程序的良好交互。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，我们必须考虑像在数值分析中那样的精度问题。计算局部域上的伽罗瓦群的主要问题是我们不能容易地获得零点及其近似，这就不允许对全局域的已知算法进行变换。我们想通过绝对伽罗华群和局部类场理论的知识来攻击伽罗华群的计算。申请人管理一个数字字段数据库，其中填充了200多万个多项式。该数据库应进行扩展，并通过具有小特征的本地功能字段进行扩展。这些数据对于寻找和理解几何和数论中有趣的猜想例子是非常重要的。此外，大表格对于做出和检验关于这类物体的渐近性的猜想很有用。

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.