On the unramified Kummer extensions of quadratic extensions of the prime cyclotomic number field (Q804629)

Let \(\ell\) be an odd prime and d be a square free rational integer, prime to \(\ell\). In this paper we give sufficient conditions to obtain the unramified Kummer extensions of \({\mathbb{Q}}(\exp (2\pi i/\ell), \sqrt{d}),\) of degree \(\ell\) or \(\ell^ 2\), using the \(\ell\)-th roots of the units of \({\mathbb{Q}}(\sqrt{(-1)^{(\ell -1)/2}\ell}, \sqrt{d}),\) when d is not a quadratic residue modulo \(\ell\). If d is a quadratic residue modulo \(\ell\), then sufficient conditions like these were already given in the previous paper [Nagoya Math. J. 115, 151-164 (1989; Zbl 0659.12007)].