zbMATH — the first resource for mathematics

Arithmetical characterizations of divisor class groups. II. (English) Zbl 0830.20084
[For part I cf. Arch. Math. 54, No. 5, 455-464 (1990; Zbl 0703.20059).]
It has been shown by D. E. Rush [Math. Proc. Camb. Philos. Soc. 94, 23-28 (1983; Zbl 0522.12005)], J. Kaczorowski [Colloq. Math. 48, 265-267 (1984; Zbl 0557.12004)] and F. Halter-Koch [Expo. Math. 8, 27-66 (1990; Zbl 0698.20054)] that the class-group of an algebraic number field $$K$$ can be described in an arithmetical way, using only factorization properties of integers of $$K$$. The author gives here a unified treatment for these proofs, based on the notion of independence in a commutative semigroup with divisor theory (as considered e.g. in F. Halter-Koch, loc. cit.) in which every divisor class contains at least one prime divisor.

MSC:
 20M14 Commutative semigroups 11R29 Class numbers, class groups, discriminants 11R27 Units and factorization 11R04 Algebraic numbers; rings of algebraic integers
Full Text: