The elementary theory of normal Frobenius fields
type: Item
instance of: scholarly article
title: The elementary theory of normal Frobenius fields
zbMATH Open document ID: 0535.12016
author: Moshe Jarden
published in: Michigan Mathematical Journal
publication date: 1983-01-01T00:00:00Z
review text: Let K be a given Hilbertian field with an elimination theory, and let M be a Frobenius field that contains K and has absolute Galois group G(M). For a profinite group G, denote by Im G the set of all finite quotient groups of G. The author of the present paper generalizes previous results of \textit{M. Fried, D. Haran} and himself [Adv. Math. 51, 1-35 (1984)] as follows. Consider a class \(\Pi\) of profinite groups, each of which appears as the absolute Galois group of a Frobenius field M. This class is supposed to be equipped with a primitive recursive algorithm to determine for given \(m+n\) finite groups \(G_ 1,...,G_ m\), \(H_ 1,...,H_ n\) whether or not there exists a \(P\in \Pi\) such that \(G_ 1,...,G_ m\in Im P\) and \(H_ 1,...,H_ n\not\in Im P\). Denoting by \({\mathcal M}\) the class of all perfect Frobenius fields M that contain K and satisfy G(M)\(\in \Pi\), it is shown that there exists a primitive recursive procedure for the theory of \({\mathcal M}.\) Using this fact and previous results of \textit{O. V. Mel'nikov} [Math. USSR, Izv. 12, 1-20 (1978); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 42, 3-25 (1978; Zbl 0382.20032)], it is shown that the theory of normal Frobenius fields that contain K is primitive recursive via Galois stratification. Finally, using a recent result of \textit{D. Haran} and \textit{A. Lubotzky} [Isr. J. Math. 41, 181-202 (1982; Zbl 0502.20013)], it follows that the theory of all perfect Frobenius fields that contain K is decidable. - The paper ends with a rich bibliography.
Mathematics Subject Classification ID: 12L05
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 03B25
Mathematics Subject Classification ID: 11S25
Mathematics Subject Classification ID: 20E18
zbMATH DE Number: 3849369
zbMATH Keywords: profinite group
zbMATH Keywords: theory of all perfect Frobenius fields is decidable
zbMATH Keywords: Hilbertian field with an elimination theory
zbMATH Keywords: absolute Galois group
zbMATH Keywords: primitive recursive algorithm
zbMATH Keywords: Galois stratification
zbMATH Keywords: bibliography
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029002845
OpenAlex ID: W2042954860
DOI: 10.1307/MMJ/1029002845
description: scientific article; zbMATH DE number 3849369