Comparison of classifications of 2-dimensional local fields, type II

Abstract

The article contributes to the theory of elimination of wild ramification for 2-dimensional local fields. It continues the study of classification of complete discrete valuation fields introduced in the work of Masato Kurihara. The main object of study is a 2-dimensional local field K of mixed characteristic with a finite residue field of odd characteristic. If such a field is weakly unramified over its constant subfield k (the maximal usual local field inside it), i. e., if eK/k = 1, its structure is well known. It is also known that any 2-dimensional local field can be turned into a standard one by means of a finite extension of its constant subfield (Epp’s theorem). However, the estimate of the minimal degree of such extension is an open question in general. In Kurihara’s article the 2-dimensional are subdivided into 2 types as follows. Consider a non-trivial linear relation between differentials of the two local parameters of the field. The field belongs to Type I, if the valuation of the coefficient by the uniformizer is less than that by the second local parameter, and to Type II otherwise. This paper is devoted to the fields of Type II. For them we consider the invariant Δ, the difference between valuations of coefficients in the above mentioned linear relation (it is non-positive for the fields of Type II). The minimal degree of the required extension of k cannot be less than eK/k for trivial reasons. However, such extension of degree eK/k does not exist in general. In this article it is proved that it exists if the absolute value of Δ is sufficiently large. The corresponding estimate for Δ depends only on eK/k.