The torsion group of a field defined by radicals
type: Item
instance of: scholarly article
title: The torsion group of a field defined by radicals
zbMATH Open document ID: 0548.12012
DOI: 10.1016/0022-314X(84)90112-4
published in: Journal of Number Theory
publication date: 1984-01-01T00:00:00Z
review text: Let K/F be a finite, abelian extension of fields, and let p be a prime which divides the degree [K:F] of the extension. Let \(K^{\times}\) (resp. \(F^{\times})\) denote the multiplicative group of nonzero elements of K (resp. F), and let \(T_ p\) denote the p-torsion subgroup of the quotient group \(K^{\times}/F^{\times}\). If K does not contain a primitive p-th root of unity, then \(T_ p\) is trivial. If K, but not F, contains a primitive p-th root of unity, then \(T_ p\) is generated by the cosets of all \(p^ i-th\) roots of unity which are contained in K. Finally, if F contains a primitive p-th root of unity, then the authors obtain a complicated result which relates the rational decomposition of \(T_ p\) to the rational decomposition of the p-torsion subgroup of the Galois group of K/F. This information is used to determine the structure of the p-torsion subgroup of \(F(\alpha)^{\times}/F^{\times}\), where \(F(\alpha)\) is a separable, radical (i.e., \(\alpha^ m\in F\) for some positive integer m) extension of F.
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 20F29
zbMATH DE Number: 3873456
zbMATH Keywords: radical extension
zbMATH Keywords: separable extension
zbMATH Keywords: Kummer theory
zbMATH Keywords: abelian extension
zbMATH Keywords: structure of the p-torsion subgroup
author: Maria T. Acosta-de-Orozco
author: William Yslas Vélez
reviewed by: H. F. Kreimer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3873456