On the Lengths of Factorizations of Elements in an Algebraic Number Ring

Abstract

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 V ( n) be the set of all integers m for which there exists irreducible elements α 1, ..., α n , β 1, ..., β m of D such that α 1 · · · α n = β 1 · · · β m . We then set Φ(n) = | V(n)| . 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].