In 2001, M. Bhargava stunned the mathematical world by extending Gauss's
200-year-old group law on integral binary quadratic forms, now familiar as the
ideal class group of a quadratic ring, to yield group laws on a vast assortment
of analogous objects. His method yields parametrizations of rings of degree up
to 5 over the integers, as well as aspects of their ideal structure, and can be
employed to yield statistical information about such rings and the associated
number fields.
In this paper, we extend a selection of Bhargava's most striking
parametrizations to cases where the base ring is not Z but an arbitrary
Dedekind domain R. We find that, once the ideal classes of R are properly
included, we readily get bijections parametrizing quadratic, cubic, and quartic
rings, as well as an analogue of the 2x2x2 cube law reinterpreting Gauss
composition for which Bhargava is famous. We expect that our results will shed
light on the analytic distribution of extensions of degree up to 4 of a fixed
number field and their ideal structure.Comment: 39 pages, 1 figure. Harvard College senior thesis, edite