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Systems of sets of lengths. (Systeme von Längenmengen.) (German) Zbl 0721.11042

The well-known result of L. Carlitz [Proc. Am. Math. Soc. 11, 391-392 (1960; Zbl 0202.33101)] states that the length of factorization into irreducibles of an algebraic integer from an algebraic number field K does not depend on the factorization if and only if the class number is 1 or 2.
W. Narkiewicz [Elementary and analytic theory of algebraic numbers (1974; Zbl 0276.12002) (2nd ed. 1990; Zbl 0717.11045)] asked about a pure arithmetical description of number fields with a given class number (Unsolved Problem No.32) or, more generally, with a given class group. This was done by the reviewer [Colloq. Math. 48, 265-267 (1984; Zbl 0557.12004)]; the solution was based on the properties of absolutely irreducible elements.
The present author tries to answer Narkiewicz’s question in “Carlitz style”, i.e. to characterize the class group in terms of lengths of factorizations. He proves among others that the set of lengths determines the class group up to a finite number of cases (theorem 2.3). The determination is unique for class groups of the form \(C_ n\), \(C^ n_ 2\), \(C_ n\oplus C_ 2\) or for class groups with small Davenport constant D (theorem 4). For \(D\leq 4\) the sets of lengths are given explicitly (theorem 3). All theorems are proved in more general setting for semigroups with divisor theory having a finite divisor class group with at least one prime divisor in each class.

MSC:

11R27 Units and factorization
11R29 Class numbers, class groups, discriminants
20M14 Commutative semigroups
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