On non-unique factorization into irreducible elements. II. (English) Zbl 0703.11057

Geroldinger, A.

Number theory. Vol. II. Diophantine and algebraic, Proc. Conf., Budapest/Hung. 1987, Colloq. Math. Soc. János Bolyai 51, 723-757 (1990).

[For the entire collection see Zbl 0694.00006.]
Let \(H\) be a semigroup with divisor theory and finite divisor class group \(G\). A most interesting example for a semigroup with divisor theory is the multiplicative semigroup of the ring of integers in an algebraic number field. Every element \(a\in H\setminus H^{\times}\) has a factorization \(a=u_ 1... u_ k\) into irreducible elements \(u_ 1,...,u_ k\); \(k\) is called length of the factorization and let \(L(a)=\{k\mid a\) has a factorization of length \(k\}\) denote the set of lengths of \(a\). In Part I [Math. Z. 197, 505-529 (1988; Zbl 0618.12002)] it was proved that sets of lengths are essentially arithmetical progressions depending only on \(G\). In this paper the possible distances of such arithmetical progressions are studied. In the case of a cyclic divisor class group a certain subset of all possible distances is determined by methods of diophantine approximation.

Reviewer: A.Geroldinger

MSC:

11R27	Units and factorization
13F05	Dedekind, Prüfer, Krull and Mori rings and their generalizations
11J70	Continued fractions and generalizations

Keywords:

multiplicative semigroup; irreducible elements; length of the factorization; cyclic divisor class group

Citations:

Zbl 0636.12002; Zbl 0694.00006; Zbl 0618.12002