zbMATH — the first resource for mathematics

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)

MSC:
 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
Full Text:
References:
 [1] Berndt B. C., Bull. Amer. Math. Soc. (N.S.) 5 (2) pp 107– (1981) · Zbl 0471.10028 [2] Fröhlich A., Invent. Math. 17 pp 143– (1972) · Zbl 0261.12008 [3] Ireland K., A classical introduction to modern number theory, (1990) · Zbl 0712.11001 [4] Louboutin S., C. R. Acad. Sci. Paris Sér. I Math. 323 (5) pp 443– (1996) [5] Louboutin S., Math. Comp. [6] Mignotte M., C. R. Acad. Sci. Paris Sér. I Math. 324 (4) pp 377– (1997) [7] Schoof R., Math. Comp. 50 (182) pp 543– (1988) [8] Schwarz W., Acta Arith. 72 (3) pp 277– (1995) [9] Seah E., Math. Comp. 41 (163) pp 303– (1983) [10] Steiner R., Math. Comp. 67 (223) pp 1317– (1998) · Zbl 0897.11009 [11] Tijdeman R., Acta Arith. 29 (2) pp 197– (1976) [12] Wall H. S., Analytic theory of continued fractions (1948) · Zbl 0035.03601 [13] Washington L. C., Introduction to cyclotomic fields,, 2. ed. (1997) · Zbl 0966.11047 [14] Williams H. C., Math. Comp. 30 (136) pp 887– (1976)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.