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Note on the class number of the \(p\)th cyclotomic field. III. (English) Zbl 1427.11114

Summary: Let \(p=2\ell^f+1\) be a prime number with \(f \geqslant 2\) and an odd prime number \(\ell\). For \(0 \leqslant t \leqslant f\), let \(K_t\) be the imaginary subfield of the \(p\)th cyclotomic field \(\mathbb{Q}(\zeta_p)\) with \([K_t : \mathbb{Q}]=2\ell^t\). Denote by \(h_{p,t}^-\) the relative class number of \(K_t\), and by \(h_{p,t}^+\) the class number of the maximal real subfield \(K_t^+\). It is known that the ratio \(h_{p,f}^-/h_{p,f-1}^-\) is odd (and hence so is \(h_{p,f}^+/h_{p,f-1}^+\)) whenever 2 is a primitive root modulo \(\ell^2\). We show that \(h_{p,f}^+/h_{p,f-1}^+\) is odd under a somewhat milder assumption on \(\ell\) and that the ratio \(h_{p,f-1}^-/h_{p,f-2}^-\) is always odd when \(\ell=3\).
For Part II, see [S. Fujima and the author, Exp. Math. 27, No. 1, 111–118 (2018; Zbl 1427.11113)].

MSC:

11R18 Cyclotomic extensions
11R29 Class numbers, class groups, discriminants

Citations:

Zbl 1427.11113
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References:

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