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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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References:

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