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On non-unique factorizations into irreducible elements. (Über nicht-eindeutige Zerlegungen in irreduzible Elemente.) (German) Zbl 0618.12002
Let $$R$$ be the ring of integers of an algebraic number field $$K$$ and let $$G$$ be the ideal class group. Every $$a\in R\setminus (R^{\times}\cup \{0\})$$ has a (not necessarily unique) factorization $$a=u_ 1\cdot...\cdot u_ k$$ into irreducible elements $$u_ 1,...,u_ k\in R$$; then $$k$$ is called the length of the factorization and $$L(a)=\{k \mid a$$ has factorization of length $$k\}$$ the set of lengths of $$a$$. Słiwa proved: if $$\# G\geq 3$$, then for every $$m\in {\mathbb N}$$ there is an element $$a\in R\setminus (R^{\times}\cup \{0\})$$ with $${\#}L(a)=m.$$
In this paper the structure of sets of lengths is investigated. The arithmetical problem for sets of lengths is translated into a combinatorial problem for the abelian group $$G$$. Let $$D(G)$$ be Davenport’s constant; then Theorem 1 asserts the existence of another constant $$M(G)$$ also depending only on $$G$$ such that every set of lengths $$L$$ has the following form: $$L=\{x_ 1,...,x_{\alpha}$$, $$y,y+{\mathfrak d}_ 1,...$$, $$y+{\mathfrak d}_{\mu}$$, $$y+d,y+{\mathfrak d}_ 1+d,...,y+{\mathfrak d}_{\mu}+d$$, $$y+2d,...$$, $$y+{\mathfrak d}_ 1+(k-1)d,...,y+{\mathfrak d}_{\mu}+(k-1)d$$, $$y+kd$$, $$z_ 1,...,z_{\beta}\}$$ with $$x_ 1<...<x_{\alpha}<y<y+{\mathfrak d}_ 1<...<y+{\mathfrak d}_{\mu}<y+d<y+kd<z_ 1<...<z_{\beta}$$, $$\alpha\leq M(G)$$, $$\beta\leq M(G)$$ and $$1\leq d\leq D(G)-2.$$ Using analytical methods it is further proved that (in the sense of density) almost all sets of lengths are as simple as possible: they are of the form $$L=\{y,y+1,...,y+k\}$$ (Theorem 2).
Reviewer: A. Geroldinger

MSC:
 11R04 Algebraic numbers; rings of algebraic integers 11R27 Units and factorization 11R23 Iwasawa theory 20K01 Finite abelian groups 20D60 Arithmetic and combinatorial problems involving abstract finite groups
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