On the first layer of anti-cyclotomic \(\mathbb Z_{p}\)-extension of imaginary quadratic fields (Q2381151)

Let \(k\) be a quadratic imaginary number field, let \(p>3\) be a prime and let \(k_1^a\) be the first layer of the anticyclotomic \(\mathbb{Z}_p\)-extension of \(k\). Under some mild hypotheses (namely \(p\) does not divide the class number of \(k\) and \(k\not\subset \mathbb{Q}(\zeta_p)\) where \(\zeta_p\) is a primitive \(p\)th root of unity) the author describes a unit \(\varepsilon\in k(\zeta_p)\) such that \(k_1^a= k(\text{Tr}_{k(\zeta_p,\varepsilon^{1/p})/ k_1^a}(\varepsilon^{1/p}))\).