New numerical invariants for totally ramified extensions of local fields
type: Item
instance of: scholarly article
title: New numerical invariants for totally ramified extensions of local fields
zbMATH Open document ID: 0876.11053
author: Volker Heiermann
published in: Journal of Number Theory
publication date: 1997-11-26T00:00:00Z
review text: The main object of this paper is to study totally ramified extensions of local fields of unequal characteristic. Such an extension \(L/K\) could be approached either via Eisenstein polynomials (Krasner) or via formal power series (Arf). For instance given a uniformizing parameter \(t\) of \(K\), given a pair \(({\mathcal F},n)\), \({\mathcal F}\in {\mathcal O}_KX^*\), \(n\in \mathbb{N}^\bullet\), there exists a unique totally ramified extension \(L/K\) which admits a uniformizing parameter \(u\) such that \(t=u^n {\mathcal F} (u)\). Using Arf's approach, the author constructs a category which is equivalent to the category of totally ramified extensions of \(K\): the objects are pairs \(({\mathcal F},n)\), and the morphisms are derived from a certain multiplication \(({\mathcal H},m)* ({\mathcal F},n)\). To study the multiplication \(*\), the author introduces new invariants attached to a totally ramified extension \(L/K\): a finite set of numerical invariants called ``inseparability indices'' (the definition of the invariants and the proof that they are indeed invariants, are very technical), and a finite set of functions \(\varphi_i\) which refine the Hasse-Herbrand function \(\varphi_{L/K}\). Using these, he proves his main result, namely the ``continuity'' of the multiplication \(*\), which allows him to recover and refine old results of Krasner and Arf.
Mathematics Subject Classification ID: 11S15
zbMATH DE Number: 913835
zbMATH Keywords: inseparability indices
zbMATH Keywords: totally ramified extensions of local fields
zbMATH Keywords: formal power series
zbMATH Keywords: Hasse-Herbrand function
reviewed by: Thong Nguyen Quang Do
MaRDI profile type: Publication
full work available at URL: jnth.1996.0092
OpenAlex ID: W2032800926
DOI: 10.1006/JNTH.1996.0092
description: scientific article; zbMATH DE number 913835