Galois Groups of Local Function Fields

计算伽罗瓦组在过去几年一直是一个非常活跃的话题。申请人对伽罗瓦群在有理数上的计算作出了许多贡献。许多年前，由于算法依赖于预先计算的数据，这些计算被限制在有限的程度上。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，因为在合适的域推广中有很好的近似根。有了这些，我们可以用Stauduhar算法的一种变体来计算根上的伽罗瓦群作为置换群。局部域上多项式的情况是完全不同的。除了在分裂域中，我们没有（近似）访问根，这使得不可能应用上述方法。在过去的几年里，一些人参与了p-adic的案子。在我的指导下，Christian Greve于2010年完成了博士论文。他利用爱森斯坦多项式的分枝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出分裂域N的子域T，使得N/T是p群扩展。这个结果给出了伽罗瓦群是p群的情形的约化。有理数的绝对伽罗瓦群仍然没有被完全理解，而局部场的绝对伽罗瓦群在一些非常罕见的情况下是已知的。如果我们限制于给定局部域的最大正反扩展，这些群在所有情况下都是已知的。在p进的情况下，这些组被有限地生成，最多有一个关系。如果单位的p根不包含在给定的域中，则这个群是一个自由的亲p群。在局部函数域情况下，极大亲-对扩展的伽罗瓦群总是自由的，但它具有可数无穷秩。在这个项目中，由于几个原因，我们将重点放在本地功能字段的案例上。首先，没有针对这种情况的实现算法（与p进的情况相比）。而且，我们觉得有些事情应该比p进的情况更简单。我们打算使用局部类场论和理论上已知的绝对伽罗瓦群的结构。给定扩展的子字段提供了关于伽罗瓦群的一些信息。申请人与Mark van Hoeij一起开发了一种计算子字段的新算法，该算法在数字字段的情况下实现。这个设置是非常通用的，因此它也应该适用于本地情况。我们相信局部函数域比p进域更容易。这个项目的另一个目标是扩展现有的数字字段数据库。在此基础上，建立了具有小特征和有界判别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.