Rings of small rank over a Dedekind domain and their ideals.

The aim of this paper is to find and prove generalizations of some of the beautiful integral parametrizations in Bhargava's theory of higher composition laws to the case where the base ring $$\mathbb {Z}$$ is replaced by an arbitrary Dedekind domain R. Specifically, we parametrize quadratic, cubic, and quartic algebras over R as well as ideal classes in quadratic algebras, getting a description of the multiplication law on ideals that extends Bhargava's famous reinterpretation of Gauss composition of binary quadratic forms. We expect that our results will shed light on the statistical properties of number field extensions of degrees 2, 3, and 4. [ABSTRACT FROM AUTHOR]