Weber's class number problem in the cyclotomic \(\mathbb Z_2\)-extension of \(\mathbb Q\). II.
Let \(\zeta_n=\exp\left(\frac{2\pi i}{2^n}\right) \), \({\mathbb Q}_n= {\mathbb Q}(\zeta_{n+2}+\zeta_{n+2}^{-1})\), \(h_n\) the class number of \({\mathbb Q}_n\). Let \(l\) be an odd prime number, \(2^c\) the exact power of \(2\) dividing \(l-1\) or \(l^2-1\) according as \(l\equiv 1\pmod 4\) or not, \(m_l=2c-3+[\log_2l] \). The authors prove that \(l\) does not divide \(\frac{h_n}{h_{m_l}} \) for any integer \(n\geq m_l\). If \(l\equiv \pm 9\pmod{16}\), then \(l\) does not divide \(h_n\) for all \(n\geq 1\). For Part I, see Exp. Math. 18, No. 2, 213--222 (2009; Zbl 1189.11033), Part III, Int. J. Number Theory 7, No. 6, 1627--1635 (2011; Zbl 1226.11119).
cyclotomic \(\mathbb Z_2\)-extension of \(\mathbb Q\)
scientific article; zbMATH DE number 5862104
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