## Abstract

This article is devoted to p-extensions of complete discrete valuation fields of mixed characteristic where p is the characteristic of the residue field. It is known that any totally ramified Galois extension with a non-maximal ramification jump can be determined by an Artin-Schreier equation, and the upper bound for the ramification jump corresponds to the lower bound of the valuation in the right-hand side of the equation. The problem of construction of extensions with arbitrary Galois groups is not solved. Inaba considered p-extensions of fields of characteristic p corresponding to a matrix equation X^{(p)} = AX herein referred to as Inaba equation. Here X^{(p)} is the result of raising each element of a square matrix X to power p, and A is a unipotent matrix over a given field. Such an equation determines a sequence of Artin-Schreier extensions. It was proved that any Inaba equation determines a Galois extension, and vice versa any finite Galois p-extension can be determined by an equation of this sort. In this article for mixed characteristic fields we prove that an extension given by an Inaba extension is a Galois extension provided that the valuations of the elements of the matrix A satisfy certain lower bounds, i. e., the ramification jumps of intermediate extensions of degree p are sufficiently small. This construction can be used in studying the field embedding problem in Galois theory. It is proved that any non-cyclic Galois extension of degree p^{2} with sufficiently small ramification jumps can be embedded into an extension with the Galois group isomorphic to the group of unipotent 3 × 3 matrices over Fp. The final part of the article contains a number of open questions that can be possibly approached by means of this construction.

Translated title of the contribution | Inaba extension of complete field of characteristic 02 |
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Original language | Russian |

Pages (from-to) | 124-133 |

Number of pages | 10 |

Journal | Chebyshevskii Sbornik |

Volume | 20 |

Issue number | 3 |

DOIs | |

State | Published - 2019 |

## Scopus subject areas

- Mathematics(all)

## Keywords

- Artin-Schreier equation
- Discrete valuation field
- Ramification jump