On the Lengths of Factorizations of Elements in an Algebraic Number Ring Original Research Article

Let D be an integral domain in which each nonzero nonunit can be written as a finite product of irreducible elements from D (such a domain is called atomic). For any positive integer n, let image(n) be the set of all integers m for which there exists irreducible elements α1, ..., αn, β1, ..., βm of D such that α1 · · · αn = β1 · · · βm. We then set image. In this paper, we consider this function and its asymptotic behavior for a large class of Dedekind domains including rings of integers of algebraic number fields). In particular we prove the following. THEOREM. Let D be a Dedekind domain with finite class group G such that every ideal class contains at least one prime ideal; let D(G) be the Davenport constant of G (see [4]), then,[formula].