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A class number problem in the cyclotomic \(\mathbb Z_3\)-extension of \(\mathbb Q\). (English) Zbl 1205.11116
Let \(\Omega_n=\mathbb Q\left(2\cos\bigl(\frac{2\pi}{3^{n+1}}\bigr)\right)\). This is the \(n\)-th lyer of the cyclotomic \(\mathbb Z_3\)-extension of \(\mathbb Q\). Let \(h_n\) be the class number of \(\Omega_n\). Let \(l\geq 5\) be a prime number and \(3^s\) the exact power of \(3\) dividing \(l^2-1\). Put \[ m_l=3s+2+[\log_3(l-1)]+\left[\log_3\frac{l-1}{2}\right]+[\log_3(2s+1+[\log_3(l-1)])], \] where \([x]\) denotes the greatest integer not exceeding \(x\). The author proves that if \(l\) does not divide \(h_{m_l}\), then \(l\) does not divide \(h_n\) for any positive integer \(n\). As a corollary, if \(l<10000\) then \(l\) does not divide \(h_n\) for any positive integer \(n\).

MSC:
11R23 Iwasawa theory
11R29 Class numbers, class groups, discriminants
11R18 Cyclotomic extensions
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