Computation of relative class numbers of imaginary abelian number fields. (English) Zbl 0929.11065

The author proposes an algorithm to compute relative class numbers of imaginary abelian number fields. The algorithm requires only \(O(\sqrt f\log f)\) elementary operations for a field of conductor \(f\). The author’s method is based on an exponentially rapidly converging series for generalized Bernoulli numbers. In addition he uses an exponentially rapidly converging series for the Artin root numbers involved.
In the last sections of his paper the author presents numerical data for fields of 2-power degree. Most of these regard fields of degree 4.
Reviewer: Rene Schoof (Roma)


11Y40 Algebraic number theory computations
11R29 Class numbers, class groups, discriminants
11R20 Other abelian and metabelian extensions
11M20 Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\)
11R42 Zeta functions and \(L\)-functions of number fields
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