The degrees of the cyclotomic extension fields
type: Item
instance of: scholarly article
title: The degrees of the cyclotomic extension fields
zbMATH Open document ID: 0802.11039
DOI: 10.1016/0024-3795(93)90263-N
author: William C. Waterhouse
published in: Linear Algebra and its Applications
publication date: 1994-12-15T00:00:00Z
review text: The following question is investigated: ``Suppose \(r\) and \(s\) are relatively prime positive integers and \(\xi_ r\), \(\xi_ s\), \(\xi_{rs}\) primitive roots of unity. When for positive integers \(a\), \(b\) and \(c\) is there a field \(K\) of characteristic zero with \(| K(\xi_{rs}): K|=a\), \(| K(\xi_ r): K| =b\), and \(| K(\xi_ s):K| =c\)?'' This problem is solved (Theorem 9) by giving four conditions on the divisibility relation among the integers \(a\), \(b\), \(c\), \(\varphi(r)\), \(\varphi(s)\), and \(g\), where \(g\) is the order of the largest group isomorphic to a subgroup of both \((\mathbb{Z}/ r\mathbb{Z})^ \times\) and \((\mathbb{Z}/ s\mathbb{Z})^ \times\). This question for fields with positive characteristic \(p\) (\(r\), \(s\) are not divisible by \(p\)) is not so complicated and is solved by Theorem 3: \(a= \text{lcm} (b,c)\). The motivation of this problem lies in the question about the period of a matrix investigated by \textit{R. B. Richter} and \textit{W. P. Wardlaw} [Linear Algebra Appl. 160, 87-97 (1992; Zbl 0755.15015)].
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 519696
zbMATH Keywords: subfields of cyclotomic extensions
reviewed by: Ladislav Skula
MaRDI profile type: Publication
cites work: Q5798081
cites work: Q3963102
cites work: Q4771385
cites work: The smallest matrix of given period and primitive roots of unity
description: scientific article; zbMATH DE number 519696