Finitely generated pro-\(p\)-groups as Galois groups of maximal \(p\)-extensions of function fields over \(\mathbb{Q}_ q\)
type: Item
instance of: scholarly article
title: Finitely generated pro-\(p\)-groups as Galois groups of maximal \(p\)-extensions of function fields over \(\mathbb{Q}_ q\)
zbMATH Open document ID: 0857.12001
DOI: 10.1007/BF02568303
author: Alexander Prestel
author: Christian U. Jensen
published in: Manuscripta Mathematica
publication date: 1997-03-11T00:00:00Z
full work available at URL: https://eudml.org/doc/156191
review text: Let \(p\) and \(q\) be two different rational primes. This article completely characterizes those (topologically) finitely generated abelian pro-\(p\)-groups which occur as Galois groups of the maximal abelian \(p\)-extension of some field of transcendence degree one over the field \(\mathbb{Q}_q\) of \(q\)-adic numbers. It is shown that the structure of these groups depends only on the algebraic properties of \(\mathbb{Q}_q\), i.e. the result is the same if \(\mathbb{Q}_q\) is replaced by any \(q\)-adically closed field. The article concludes with explicit constructions of fields realizing the groups in question.
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 12F12
zbMATH DE Number: 922475
zbMATH Keywords: maximal abelian \(p\)-extension
zbMATH Keywords: \(q\)-adically closed field
zbMATH Keywords: finitely generated abelian pro-\(p\)-groups
zbMATH Keywords: Galois groups
zbMATH Keywords: transcendence degree one
MaRDI profile type: Publication
cites work: Q4405003
cites work: Galois groups of intersections of local fields
cites work: On finite intersections of ''Henselian valued'' fields
cites work: Realization of finitely generated profinite groups by maximal abelian extensions of fields.
cites work: Freie Produkte pro-endlicher Gruppen und ihre Kohomologie
cites work: Galoissche Kennzeichnung p-adisch abgeschlossener Körper.
cites work: On Grothendieck's conjecture of birational anabelian geometry
cites work: Generic Galois extensions and problems in field theory
cites work: Profinite Groups, Arithmetic, and Geometry. (AM-67)
OpenAlex ID: W2067859264
description: scientific article; zbMATH DE number 922475