Inaba extension of complete field of characteristic~$0$

S. V. Vostokov ; I. B. Zhukov ; O. Yu. Ivanova

Čebyševskij sbornik, Tome 21 (2020) no. 3, pp. 59-67

Résumé

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.

Détail

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