An Analysis Using the Zaks-Skula Constant of Element Factorizations in Dedekind Domains Original Research Article

Let D be a Dedekind domain with finite class group G. Let α be a nonzero nonunit of D and suppose β1, ..., βs, γ1, ..., γt are irreducibles of D such that α = β1 ··· βs = γ1 ··· γt. Two recent papers (J. Steffan, 1986, J. Algebra102, 229-236; R. J. Valenza, 1990, J. Number Theory38, 212-218) have studied the quotient s/t and both conclude that given D there exists a smallest real number ρ(D) such that s/t ≤ ρ(D). While it is easy to show that ρ(D) ≤ G/2, we use the Zaks-Skula constant to find a sharper upper bound on ρ(D). We then use this upper bound to explore the behavior of the Φ-function, which counts the number of different lengths of certain products, on two specific classes of Dedekind domains.