Abelian local \(p\)-class field theory
type: Item
instance of: scholarly article
title: Abelian local \(p\)-class field theory
zbMATH Open document ID: 0819.11057
DOI: 10.1007/BF01446646
published in: Mathematische Annalen
publication date: 1995-08-31T00:00:00Z
full work available at URL: https://eudml.org/doc/165308
review text: In the first section an abstract \(p\)-class field theory is developed. Assume that the ground field \(K\) possesses a distinguished Galois extension \(\widetilde {K}/K\) with the Galois group being an abelian pro- \(p\)-group. Then for a finite separable extension \(F/K\) Galois \(p\)- extensions \(L/F\) linearly disjoint with \(\widetilde {F}/F\), \(\widetilde {F}= F\widetilde {K}\) (these extensions are said to be totally ramified) are described by means of a suitable modulation \({\mathcal A}= \{A_ F\): \(F/K\}\). One can introduce a map \(\Upsilon_{L/F}\) acting from the group \(\text{Gal} (L/F)^ \sim\), of continuous \(\mathbb{Z}_ p\)-homomorphisms from \(\text{Gal} (\widetilde {F}/ F)\) to the discrete group \(\text{Gal} (L/F)\), to the quotient group \(U_ F/ N_{L/F} U_ L\), where \(U_ F\) is the kernel of the substantial epimorphism \(v: A_ F\to \mathbb{Z}\). This map is a generalization of the Neukirch map in the classical cases. One of the main conditions on which the theory rests is Hilbert Satz 90 for \(A_ F\). At the same time several other reasonable properties, such as \(\text{Gal} (L/F)\)-stable elements of \(A_ L\) are in one-to-one correspondence with elements of \(A_ F\), are not assumed to hold in general. Thereby, first of all we consider extensions for which a weaker condition: \[ v_{\widetilde {L}} (A_{\widetilde {L}}^{\text{Gal} (\widetilde {L}/\widetilde {F})})= | \widetilde{L} : \widetilde{F}\mid \mathbb{Z} \] holds for a cyclic \(p\)-extension \(\widetilde {L}/ \widetilde {F}\). The class of such extensions (they are called marked) is required to be sufficiently large. For a totally ramified Galois extension \(L/F\), such that any of its intermediate cyclic subextensions \(L_ 1/ F_ 1\) \((F\subset F_ 1\subset L_ 1\subset L)\) is marked, one can define a homomorphism \(\Psi_{L/F}\) (as a generalization of the Hazewinkel homomorphism in one-dimensional case) acting from \(U_ F/ N_{L/F} U_ L\) to \((\text{Gal} (L/F)^{\text{ab}} )^ \sim\); the…
Mathematics Subject Classification ID: 11S31
Mathematics Subject Classification ID: 19F05
Mathematics Subject Classification ID: 19D45
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11S70
Mathematics Subject Classification ID: 11S15
zbMATH DE Number: 759284
zbMATH Keywords: abelian totally ramified \(p\)-extensions
zbMATH Keywords: abstract \(p\)-class field theory
zbMATH Keywords: Galois \(p\)-extensions
zbMATH Keywords: generalization of the Neukirch map
zbMATH Keywords: Hilbert Satz 90
zbMATH Keywords: generalization of the Hazewinkel homomorphism
zbMATH Keywords: complete discrete valuation field
zbMATH Keywords: Milnor topological \(K\)-group
zbMATH Keywords: Artin-Schreier extensions
zbMATH Keywords: elementary proof of \(p\)-part of class field theory
zbMATH Keywords: higher dimensional local fields
zbMATH Keywords: ramification
zbMATH Keywords: imperfect residue field
zbMATH Keywords: Hasse-Herbrand function
zbMATH Keywords: norm subgroups
Wikidata QID: Q56688359
author: I. B. Fesenko
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 759284