Weil numbers and CM fields. II (Q1814420)

An algebraic integer \(\alpha\) is called a Weil \(n\)-number, \(n\in\mathbb{N}\), if \(|\alpha^ \sigma|^ 2=n\), for all \(\sigma\in\hbox{Aut} (\mathbb{C})\), while no \(\alpha^ \sigma\) is real. The object of this paper is to find quantitative results on the distribution of Weil \(n\)-numbers in a given CM-field \(K\). This is a sequel to a previous paper [the author and \textit{A. Greaves}, J. Reine Angew. Math. 391, 198-212 (1988; Zbl 0656.12002)] in which the special case \(n\) a prime power was considered. The author gives an asymptotic formula for the number of \(n\) which lie in the interval \([1,x]\), as \(x\to\infty\), such that \(K=\mathbb{Q}(\alpha)\), where \(\alpha\) is a Weil \(n\)-number.