
Inaba extension of complete field of characteristic $0$
S. V. Vostokov^{a}, I. B. Zhukov^{a}, O. Yu. Ivanova^{b} ^{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 nonmaximal ramification jump can be determined by an ArtinSchreier equation, and the upper bound for the ramification jump corresponds to the lower bound of the valuation in the righthand 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 ArtinSchreier 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 noncyclic 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, ArtinSchreier equation.
Funding Agency 
Grant Number 
Russian Science Foundation 
161110200 
The study was supported by the Russian science Foundation (project 161110200). 
DOI:
https://doi.org/10.22405/222683832018203124133
Full text:
PDF file (647 kB)
UDC:
512.623 Received: 04.10.2019 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 124133
\mathnet{http://mi.mathnet.ru/cheb802}
\crossref{https://doi.org/10.22405/222683832018203124133}
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http://mi.mathnet.ru/eng/cheb802 http://mi.mathnet.ru/eng/cheb/v20/i3/p124
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