The degrees of the cyclotomic extension fields (Q1316181)

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)].