TY - JOUR

T1 - Расширения Инабы полных полей характеристики 0

AU - Vostokov, S. V.

AU - Zhukov, I. B.

AU - Ivanova, O. Y.

N1 - Funding Information: 2The study was supported by the Russian science Foundation (project 16-11-10200). Publisher Copyright: © 2019 State Lev Tolstoy Pedagogical University. All rights reserved. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2019

Y1 - 2019

N2 - 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 p2 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.

AB - 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 p2 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.

KW - Artin-Schreier equation

KW - Discrete valuation field

KW - Ramification jump

UR - http://www.scopus.com/inward/record.url?scp=85079749762&partnerID=8YFLogxK

U2 - 10.22405/2226-8383-2019-20-3-124-133

DO - 10.22405/2226-8383-2019-20-3-124-133

M3 - статья

AN - SCOPUS:85079749762

VL - 20

SP - 124

EP - 133

JO - Chebyshevskii Sbornik

JF - Chebyshevskii Sbornik

SN - 2226-8383

IS - 3

ER -