Calculations in generalised lubin - Tate theory

In this paper, we study different extensions of local fields. For an arbitrary finite extension of the field of p-adic numbers $K/Q_p$ it is possible to describe, using the famous Lubin - Tate theory, its maximal abelian extension $K^{ab}/K$ 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 $T_m$ 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 $(T_m)^{ab}/K$. The main result of the paper is explicit description of the Galois group of the extension $(K^{ur})^ {ab}/K$, where $K^{ur}$ is the maximal unramified extension of the field $K$. We also applied similar methods to the study of ramified extensions of $K$.