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An application of the \(p\)-adic class number formula. (English) Zbl 0886.11061

The author proves some results about \(p\)-divisibility of \(h(K)\), the class number of a real abelian field \(K\), using Leopoldt’s class number formula in terms of \(p\)-adic \(L\)-functions. He generalizes a theorem of S. Jakubec [Abh. Math. Semin. Univ. Hamb. 63, 67-86 (1993; Zbl 0788.11052)] to \[ \prod_\chi\sum_j b_j\chi(j)\equiv 0\pmod p \] if \(p|h(K)\) (where \(\chi\neq 1\) runs over Dirichlet characters of \(K\)), by giving a variant of Leopoldt’s results from 1959, that is \(\prod_\chi B_\chi^{(p-1)}\equiv 0\pmod p\) if \(p|h(K)\), with \(B_\chi^{(p-1)}\) the \((p-1)\)st generalized Bernoulli number; and then he shows that \(p\nmid h(K)\) for \(K=\mathbb{Q}(\zeta_q)^+\) under certain conditions.

MSC:

11R29 Class numbers, class groups, discriminants
11S31 Class field theory; \(p\)-adic formal groups
11R42 Zeta functions and \(L\)-functions of number fields
11S40 Zeta functions and \(L\)-functions
11R18 Cyclotomic extensions
11R20 Other abelian and metabelian extensions

Citations:

Zbl 0788.11052
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References:

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