Abstract

In this paper, various extensions of local fields are considered. For arbitrary finite extension K of the field of p-adic numbers, the maximum Abelian extension K^Ab/K and the corresponding Galois group can be described using the well-known Lubin–Tate theory. It is represented as a direct product of groups obtained using the maximum unramified extension of K and a fully ramified extension obtained using the roots of some endomorphisms of Lubin–Tate formal groups. We consider the so-called “generalized Lubin–Tate formal groups” and extensions obtained by adding the roots of their endomorphisms to the field under consideration. Using the fact that a correctly chosen generalized formal group coincides with the classical one over unramified finite extension T_m of degree m of field K, it was possible to obtain the Galois group of the extension (T_m)^Ab/K. The main result of the work, is an explicit description of the Galois group of the extension (K^ur)^Ab/K, where K^ur is the maximum unramified extension of K. Similar methods are also used to study ramified extensions of the field K.

中文翻译：

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广义鲁宾-泰特理论中的计算

摘要

在本文中，考虑了局部字段的各种扩展。对于p- adic数域的任意有限扩展K，可以使用众所周知的Lubin-Tate理论描述最大阿贝尔扩展K ^Ab / K和相应的Galois群。它表示为使用K的最大未分叉扩展名获得的组的直接乘积并使用Lubin–Tate形式群的一些内同态的根获得了完全分支的扩展。我们考虑所谓的“广义Lubin-Tate形式族”和通过将其内同型的根加到所考虑的领域而获得的扩展。利用一个正确选择的广义形式群与经典形式一致的事实，该形式群与场K的度m的未分叉有限扩展T _m吻合，就有可能得到扩展（T _m）^Ab / K的Galois群。这项工作的主要结果是对扩展名（K ^ur）^Ab的Galois组的明确描述。/ ķ，其中ķ^乌尔是最大unramified扩展ķ。类似的方法也用于研究场K的分支扩展。