Inaba extension of complete field of characteristic 0

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.