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On divisibility of the class number of real octic fields of a prime conductor \(p=n^ 4+16\) by \(p\). (English) Zbl 0818.11042
Let \(K\) be a real octic subfield of the \(p\)-th cyclotomic field, where \(p\) is a prime of the form \(n^ 4+16\). The author proves that \(p\) divides the class number of \(K\) if and only if \(p\) divides the numerator of at least one of the Bernoulli numbers \(B_{j(p- 1)/8}\) for \(j=1,3,5,7\). In a previous work [Abh. Math. Semin. Univ. Hamb. 64, 105- 124 (1994)] he used a similar method to obtain an analogous result for quintic fields.
MSC:
11R29 Class numbers, class groups, discriminants
11R18 Cyclotomic extensions
11B68 Bernoulli and Euler numbers and polynomials
11R20 Other abelian and metabelian extensions
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