## Abstract

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.

Original language | English |
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Pages (from-to) | 131-135 |

Number of pages | 5 |

Journal | Vestnik St. Petersburg University: Mathematics |

Volume | 53 |

Issue number | 2 |

DOIs | |

State | Published - 1 Apr 2020 |

## Scopus subject areas

- Mathematics(all)

## Keywords

- formal group laws
- maximum unramified extension