Genus fields of abelian extensions of rational congruence function fields. II
type: Item
instance of: scholarly article
title: Genus fields of abelian extensions of rational congruence function fields. II
zbMATH Open document ID: 1425.11184
DOI: 10.1216/RMJ-2018-48-7-2099
author: Jonny Fernando Barreto-Castañeda
author: Carlos Montelongo-Vázquez
author: Carlos Daniel Reyes-Morales
author: Martha Rzedowski-Calderón
author: Gabriel D. Villa Salvador
published in: Rocky Mountain Journal of Mathematics
publication date: 2019-01-10T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1544756803
review text: This paper extends and improves on Part I of this paper [Finite Fields Appl. 20, 40--54 (2013; Zbl 1285.11140); corrigendum ibid. 33, 283--285 (2015; Zbl 1368.11120)] to ``compute'' the genus field of an abelian extension of rational congruence function fields. The paper's well-written introduction explains the history and evolution in terms of class field theory. Let $K$ be a global field (i.e., either a finite extension of $\mathbb{Q}$ or a finite extension of $\mathbb{F}(T)$, where $\mathbb{F}$ is a finite field and $T$ is an indeterminate). Whereas the definition of the genus field of $K$ in the number field case has been known since the work of Fröhlich in the 1950s, the function field case is a more recent development. In case $K/\mathbb{Q}$ is a finite extension, the genus field $K_g$ of $K$ is defined to be the composite field $Kk^*$, where $K^*/\mathbb{Q}$ is a maximal abelian number field such that $Kk^*/K$ is unramified. The definition in the function field case is similar but more technical. The main result, Theorem 6.9 which computes $K_g$ as a composite of simpler field extensions, is also too technical to be presented in a short review.
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R60
zbMATH DE Number: 6999256
zbMATH Keywords: global function fields
zbMATH Keywords: ramification
zbMATH Keywords: genus fields
zbMATH Keywords: abelian \(p\)-extensions
reviewed by: David Joyner
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6999256