Regular split embedding problems over function fields of one variable over ample fields
type: Item
instance of: scholarly article
title: Regular split embedding problems over function fields of one variable over ample fields
zbMATH Open document ID: 0923.12005
author: Dan Haran
author: Moshe Jarden
published in: Journal of Algebra
publication date: 1999-07-26T00:00:00Z
review text: The main result of the paper is the following: if \(K_0\) is an ample field, \(E_0\) is a function field of one variable over \(K_0\), and \(E\) is a finite Galois extension of \(E_0\), then each split (regular) \(K_0\)-embedding problem \[ G\rtimes G(E/E_0) \longrightarrow G(E/E_0) \] has a (regular) solution. This is a generalization of a theorem given by \textit{F. Pop} in an unpublished paper [The geometric case of a conjecture of Shafarevich, preprint, Heidelberg (1993)]. The paper is a continuation of a paper of the same two authors [Forum Math. 10, No. 3, 329-351 (1998; Zbl 0903.12003)], where they introduced a purely algebraic approach to embedding problems, using the ``patching principle''. The result stated above is obtained by a sequel of reduction steps: first, one can assume \(E_0=K_0(x)\), then \(K_0\) is replaced by its field of formal power series, so that it is complete under an ultrametric absolute value, and one can directly apply the patching principle.
reviewed by: Leonardo Cangelmi
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 1221659
zbMATH Keywords: inverse Galois theory
zbMATH Keywords: split embedding problems
zbMATH Keywords: regular embedding problems
zbMATH Keywords: ample fields
zbMATH Keywords: function fields
zbMATH Keywords: patching principle
MaRDI profile type: Publication
OpenAlex ID: W1999641297
cites work: Q5590141
cites work: Q3027108
cites work: Galois groups over complete valued fields
cites work: Embedding problems over large fields
cites work: Étale Galois covers of affine smooth curves. The geometric case of a conjecture of Shafarevich. On Abhyankar's conjecture
DOI: 10.1006/JABR.1998.7454
description: scientific article; zbMATH DE number 1221659