On two approaches to classification of higher local fields

O. Ivanova, S. Vostokov, I. Zhukov

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Abstract

This article links Kurihara's classification of complete discrete valuation fields and Epp's theory of elimination of wild ramification.
For any complete discrete valuation field K with arbitrary residue field of prime characteristic one can define a certain numerical invariant Γ(K) which underlies Kurihara's classification of such fields into 2 types: the field K is of Type I if and only if Γ(K) is positive. The value of this invariant indicates how distant is the given field from a standard one, i.e., from a field which is unramified over its constant subfield k which is the maximal subfield with perfect residue field. (Standard 2-dimensional local fields are exactly fields of the form k{{t}}.)
We prove (under some mild restriction on K) that for a Type I mixed characteristic 2-dimensional local field K there exists an estimate from below for [l:k] where l/k is an extension such that lK is a standard field (existing due to Epp's theory); the logarithm of this degree can be estimated linearly in terms of Γ(K) with the coefficient depending only on eK/k.
Original languageEnglish
Pages (from-to)186-197
JournalChebyshevskii Sbornik
Volume20
Issue number2
StatePublished - 2019

Keywords

  • Higher local fields
  • wild ramification

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