On non-unique factorizations into irreducible elements
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).
scientific article; zbMATH DE number 4002213