Galois Groups of Local Function Fields

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• 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; 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.

在过去几年中，计算Galois组一直是一个非常活跃的话题。申请人为对理由的Galois组计算做出了许多贡献。许多年前，这些计算已限制为有限程度，因为算法取决于预先计算的数据。如今，有一些实现可以任意地工作。这些方法或多或少可以扩展到特征0中的全局字段和函数字段，因为在可用的合适的场扩展中，根部具有良好的近似值。有了这些，我们可以通过Stauduhar's Algorith的一种变体将Galois组作为根源上的置换组。在本地领域上的多项式案例完全不同。除了在分裂字段中，我们无法（近似）对根的访问，这使得无法应用上述方法。在过去的几年中，有些人在P-ADIC案件中工作。在我的监督下，克里斯蒂安·格雷夫（Christian Greve）在2010年完成了博士学位论文。他使用艾森斯坦多项式的所谓的后果多边形迈出了非常重要的一步。对于Eisenstein多项式，他可以在多项式时间计算分裂场n的子场T，使得N/T是P组扩展。该结果减少了我们的Galois组为P组的情况。理性的绝对Galois组仍不完全了解，而在某些非常罕见的情况下，众所周知，当地田间的绝对Galois组。如果我们仅限于给定局部领域的最大pro-p延伸，则这些组在所有情况下都是已知的。在P-ADIC的情况下，这些组最多有限地生成。如果统一根源不包含在给定的字段中，则该组是一个免费的Pro-P组。在本地函数字段中，最大pro-p-extension的Galois组始终是免费的，但是它具有无限的等级。在该项目中，我们出于多种原因将重点放在本地功能字段案例上。首先，对于这种情况，没有实施算法（与P-ADIC案例相比）。此外，我们感觉到某些事情应该比P-Adic情况更容易。我们打算使用本地阶级字段理论和理论上已知的绝对Galois组的结构。给定扩展的subfields提供了有关Galois组的一些信息。申请人与马克·范·霍伊（Mark van Hoeij）一起开发了一种用于计算子场的新算法，该算法在数字字段案例中实现。该设置非常通用，因此它也应该在当地情况下工作。我们认为，本地功能字段比P-ADIC字段更容易。该项目的另一个目标是扩展数字字段的现有数据库。此外，我们想在小特征和有限的歧视范围内创建一个本地函数字段的数据库。