Invitation to higher local fields, Part II, section 7: Recovering higher global and local fields from Galois groups - an algebraic approach

2000/12/18 by Ido Efrat
Mathematics · #math.NT #math.AG #msc:12E30 #msc:12J25 #msc:19M05

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published as Geom. Topol. Monogr. Volume 3(2000) 273-279 · For introduction and notation, see math.NT/0012131 . Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon3/m3-II-7.abs.html

arxiv created 2000/12/18 · arxiv updated 2009/11/30

Abstract

A main problem in Galois theory is to characterize the fields with a given absolute Galois group. We apply a K-theoretic method for constructing valuations to study this problem in various situations. As a first application we obtain an algebraic proof of the 0-dimensional case of Grothendieck's anabelian conjecture (proven by Pop), which says that finitely generated infinite fields are determined up to purely inseparable extensions by their absolute Galois groups. As a second application (which is a joint work with Fesenko) we analyze the arithmetic structure of fields with the same absolute Galois group as a higher-dimensional local field.

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