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 Chebyshevskii Sb., 2019, Volume 20, Issue 3, Pages 124–133 (Mi cheb802)

Inaba extension of complete field of characteristic $0$

S. V. Vostokova, I. B. Zhukova, O. Yu. Ivanovab

a Saint Petersburg State University (St. Petersburg)
b Saint Petersburg State University of Aerospace Instrumentation (St. Petersburg)

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 $3\times 3$ matrices over $\mathbb F_p$.
The final part of the article contains a number of open questions that can be possibly approached by means of this construction.

Keywords: discrete valuation field, ramification jump, Artin-Schreier equation.

 Funding Agency Grant Number Russian Science Foundation 16-11-10200 The study was supported by the Russian science Foundation (project 16-11-10200).

DOI: https://doi.org/10.22405/2226-8383-2018-20-3-124-133

Full text: PDF file (647 kB)

UDC: 512.623
Accepted:12.11.2019

Citation: S. V. Vostokov, I. B. Zhukov, O. Yu. Ivanova, “Inaba extension of complete field of characteristic $0$”, Chebyshevskii Sb., 20:3 (2019), 124–133

Citation in format AMSBIB
\Bibitem{VosZhuIva19} \by S.~V.~Vostokov, I.~B.~Zhukov, O.~Yu.~Ivanova \paper Inaba extension of complete field of characteristic~$0$ \jour Chebyshevskii Sb. \yr 2019 \vol 20 \issue 3 \pages 124--133 \mathnet{http://mi.mathnet.ru/cheb802} \crossref{https://doi.org/10.22405/2226-8383-2018-20-3-124-133}