Galois Groups of Local Function Fields

计算伽罗瓦群是近年来一个非常活跃的课题。申请人已经作出了许多贡献的情况下，伽罗瓦群计算的有理数。许多年前，这些计算已被限制到有界的程度，因为算法一直依赖于预先计算的数据。现在，有一些实现可以工作于任意程度。这些方法可以或多或少地扩展到全局域和特征为0的函数域上的多项式，因为在适当的域扩展中可以很好地近似根。有了这些，我们可以通过Stauduhar算法的一个变体来计算作为根上的置换群的伽罗瓦群。局部域上多项式的情况完全不同。除了在分裂域中，我们不能（近似地）接近根，这使得不可能应用上述方法。在过去的几年里，一些人在p-adic案件中工作。在我的指导下，Christian Greve于2010年完成了他的博士论文。他提出了一个非常重要的一步，利用所谓的分歧多边形的爱森斯坦多项式。对于爱森斯坦多项式，他可以计算在多项式时间的一个子域T的分裂领域N，使N/T是一个p群的扩展。这个结果给出了一个减少的情况下，我们的伽罗瓦群是p-groups.The绝对伽罗瓦群的有理数仍然没有完全理解，而绝对伽罗瓦群的局部域是已知的一些非常罕见的情况。如果我们限制到给定局部域的最大pro-p-扩张，则这些群在所有情况下都是已知的。在p-adic的情况下，这些群是由至多一个关系生成的。如果单位的p次根不包含在给定的域中，则该群是自由亲p群。在局部函数域情形下，极大pro-p-扩张的伽罗瓦群总是自由的，但它具有可数无限秩。首先，对于这种情况没有实现的算法（与p-adic情况相比）。此外，我们觉得有些事情应该比p-adic的情况更容易。我们打算使用局部类域理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了关于伽罗瓦群的一些信息。申请人与Mark货车Hoeij一起开发了一种用于计算子域的新算法，该算法在数域情况下实现。该设置是相当一般的，因此它也应该在本地情况下工作。我们相信局部函数域比p-adic域更容易。这个项目的另一个目标是扩展现有的数域数据库。此外，我们想建立一个数据库的局部功能领域的小特征和有界的判别。

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.