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- Title
Rings of small rank over a Dedekind domain and their ideals.
- Authors
O'Dorney, Evan
- Abstract
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.
- Subjects
RING extensions (Algebra); DEDEKIND sums; PARTITIONS (Mathematics); GAUSSIAN sums; INTEGRAL domains
- Publication
Research in the Mathematical Sciences, 2016, Vol 3, Issue 1, p1
- ISSN
2522-0144
- Publication type
Article
- DOI
10.1186/s40687-016-0054-0