Weakly factorial quadratic orders

Abstract EN

In a weakly factorial domain, every nonunit element is a product of primary elements.

Weakly factorial orders are characterized by means of their group of units.

If R is a weakly factorial quadratic order, the fundamental unit of the integral closure of R determines the form of atoms of R, thereby allowing to compute the following factorization functions on R.

We denote respectively by l(x) and L(x) the inf and sup of the lengths of factorizations of a nonzero nonunit x & R into a product of irreducible elements.

Explicit formulas for l(x) and L{x) are given, from which the asymptotic behavior of these functions and the elasticity of R are deduced.