Galois Groups of Local Function Fields

计算伽罗瓦群在过去的几年里一直是一个非常活跃的话题。申请人为伽罗华群计算的有理数情形作出了许多贡献。许多年前，这些计算一直被限制在有界度上，因为算法一直依赖于预计算数据。现在，有一些实现可以在任意程度上工作。这些方法可以或多或少地推广到特征为0的全局域和函数域上的多项式，因为在适当的域扩张中存在对根的良好逼近。有了这些，我们就可以用Stauduhar算法的一个变体来计算Galois群作为根上的置换群。局部域上的多项式的情况完全不同。除了分裂域外，我们不能(近似地)访问根，这使得应用上述方法是不可能的。在过去的几年里，一些人在P-ADY案件中工作。在我的指导下，克里斯蒂安·格雷夫于2010年完成了博士论文。他利用艾森斯坦多项式的所谓分支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内计算分裂域N的一个子域T，使得N/T是p-群扩张。这一结果减少了有理数的绝对伽罗华群是p-群的情况，而局部域的绝对伽罗华群直到很少的情况下才被知道。如果我们限制到给定局部域的最大prop-扩张，则这些群在所有情况下都是已知的。在p-进的情况下，这些群是用至多一个关系有限地生成的。如果单位的p次根不包含在给定域中，则群是自由的PRO-P群。在局部函数域的情况下，极大Prop-扩张的Galois群总是自由的，但它有可数无限的秩.在这个项目中，我们主要关注局部函数域的情形.首先，没有针对这种情况的已实现的算法(与P-ADY的情况相比)。此外，我们有一种感觉，有些事情应该比p-addy情况下更容易。我们打算利用局部类场理论和理论上已知的绝对Galois群的结构。给定扩张的子域提供了关于Galois群的一些信息。申请人与Mark van Hoeij一起开发了一种计算子域的新算法，并在数域情况下实现了该算法。这种设置是非常通用的，因此它也应该适用于当地的情况。我们认为局部函数域比p进数域更容易。本项目的另一个目标是扩展现有的数值域的数据库。此外，我们还希望创建一个小特征和有界判别式的局部函数域数据库。

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.