Calculations in generalised Lubin—Tate theory

Abstract

In this paper, we study different extensions of local fields. For an arbitrary finite extension of the field of p-adic numbers K/Qp it is possible to describe, using the famous Lubin—Tate theory, its maximal abelian extension Kab/K and the corresponding Galois group. It is a Cartesian product of the groups appearing from the maximal unramified extension of K and a fully ramified extension obtained using the roots of some endomorphisms of the Lubin— Tate formal groups. Here, we are going to consider so-called generalised Lubin—Tate formal groups and the extensions that appear after adding the roots of their isomorphisms to the initial field. Using the fact that for a finite unramified extension Tm of degree m of the field K one of such formal groups coincides with a classical one, it became possible to obtain the Galois group of the extension (Tm)ab/K. The main result of the paper is explicit description of the Galois group of the extension (Kur)ab/K, where Kur is the maximal unramified extension of the field K. We also applied similar methods to the study of ramified extensions of K.