Galois Groups of Local Function Fields

局部函数域伽罗瓦群

• 批准号: 413234620
• 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: Research Grants
• 财政年份: 2018
• 资助国家: 德国
• 起止时间: 2017-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/413234620?language=en
• 关键词: Galois; Groups; Local; Function; Fields

Computing Galois groups has been a very active topic in the last years. The applicant has made many contributions for the case of Galois group computation over the rationals. Many years ago these computations have been restricted to bounded degree because the algorithms have been dependent on precomputed data. Nowadays, there are implementations which can work for arbitrary degree. These methods can more or less be extended to polynomials over global fields and function fields in characteristic 0 because there are good approximations to the roots in a suitable field extension available. With these we can compute the Galois group as a permutation group on the roots by a variant of Stauduhar's algorithm.The case of polynomials over local fields is completely different. Except in the splitting field, we do not get (approximative) access to the roots, which makes it impossible to apply the above mentioned methods. During the last years some people worked in the p-adic case. Under my supervision Christian Greve finished his PhD-thesis in 2010. He made a very important step by using the so-called ramification polygon of an Eisenstein polynomial. For Eisenstein polynomials he can compute in polynomial time a subfield T of the splitting field N such that N/T is a p-group extension. This result gives a reduction to the situation that our Galois groups are p-groups.The absolute Galois group of the rationals is still not completely understood whereas the absolute Galois group of a local field is known up to some very rare cases. If we restrict to the maximal pro-p-extension of a given local field, these groups are known in all cases. In the p-adic case these groups are finitely generated with at most one relation. If the p-th roots of unity are not contained in the given field, the group is a free pro-p group. In the local function field case the Galois group of the maximal pro-p-extension is always free, but it has countably infinite rank.In this project we focus on the local function field case for several reasons. First, there are no implemented algorithm for this situation (compared to the p-adic case). Moreover, we have the feeling that some things should be easier than in the p-adic case. We intend to use local class field theory and the structure of the theoretically known absolute Galois group.Subfields of the given extension provide some information about the Galois group. Together with Mark van Hoeij the applicant developed a new algorithm for computing subfield which was implemented in the number field case. The setup is quite general and therefore it should also work in the local situation. We believe that local function fields are easier than p-adic fields.Another goal of this project is to extend the existing database for number fields. Furthermore, we would like to create a database of local function fields in small characteristic and bounded discriminant.

计算伽罗瓦群在过去几年中一直是一个非常活跃的话题。申请人对有理数上的伽罗瓦群计算案例做出了许多贡献。许多年前，这些计算受到有限程度的限制，因为算法依赖于预先计算的数据。如今，有一些可以在任意程度上工作的实现。这些方法或多或少可以扩展到全局域和特征 0 函数域上的多项式，因为在可用的合适域扩展中存在对根的良好近似。有了这些，我们可以通过 Stauduhar 算法的变体将伽罗瓦群计算为根上的置换群。局部域上的多项式的情况完全不同。除了在分裂领域之外，我们无法（近似）访问根，这使得不可能应用上述方法。在过去的几年里，有些人从事 p-adic 案例。在我的指导下，Christian Greve 于 2010 年完成了他的博士论文。他通过使用所谓的爱森斯坦多项式的分支多边形迈出了非常重要的一步。对于爱森斯坦多项式，他可以在多项式时间内计算分裂域 N 的子域 T，使得 N/T 是 p 群扩展。这个结果简化了我们的伽罗瓦群是p群的情况。有理数的绝对伽罗瓦群仍然没有完全被理解，而局部域的绝对伽罗瓦群在一些非常罕见的情况下是已知的。如果我们限制给定局部场的最大 pro-p-extension，则这些组在所有情况下都是已知的。在 p-adic 情况下，这些群是有限生成的，最多具有一个关系。如果给定域中不包含 p 次单位根，则该群是自由亲 p 群。在局部函数域的情况下，最大 pro-p-扩张的伽罗瓦群总是自由的，但它具有可数无限的秩。在这个项目中，我们出于几个原因关注局部函数域的情况。首先，没有针对这种情况的实现算法（与 p-adic 情况相比）。此外，我们感觉有些事情应该比 p-adic 情况更容易。我们打算使用局部类域论和理论上已知的绝对伽罗瓦群的结构。给定扩展的子域提供有关伽罗瓦群的一些信息。申请人与Mark van Hoeij 一起开发了一种用于计算子域的新算法，该算法已在数域案例中实现。该设置非常通用，因此它也应该适用于本地情况。我们相信局部函数域比 p-adic 域更容易。该项目的另一个目标是扩展数字域的现有数据库。此外，我们希望创建一个小特征和有界判别式局部函数域的数据库。