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Relative class numbers inside the \(p\)th cyclotomic field. (English) Zbl 1458.11157

Any prime number \(p\equiv 3\pmod{4}\) can be written (not uniquely) in the form \(p=2nl^f+1\) for some odd \(n\) and prime \(l\) with \(l\nmid n\). Now, for every \(0\leq t\leq f\) we can define \(K_t\) the imaginary subfield of \(\mathbb{Q}(\zeta_p)\) of degree \(t\) and let \(h_t^{-}\) the relative class number of \(K_t\). In this paper, the authors give some divisibility results about the the ratio \(h_t^{-}/h_{t-1}^{-}\).

MSC:

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

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Full Text: Euclid

References:

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