  The association of algebraic objects to forms has had many important
applications in number theory. Gauss, over two centuries ago, studied quadratic
rings and ideals associated to binary quadratic forms, and found that ideal
classes of quadratic rings are exactly parametrized by equivalence classes of
integral binary quadratic forms. Delone and Faddeev, in 1940, showed that cubic
rings are parametrized by equivalence classes of integral binary cubic forms.
Birch, Merriman, Nakagawa, Corso, Dvornicich, and Simon have all studied rings
associated to binary forms of degree n for any n, but it has not previously
been known which rings, and with what additional structure, are associated to
binary forms. In this paper, we show exactly what algebraic structures are
parametrized by binary n-ic forms, for all n. The algebraic data associated to
an integral binary n-ic form includes a ring isomorphic to $\mathbb{Z}^n$ as a
$\mathbb{Z}$-module, an ideal class for that ring, and a condition on the ring
and ideal class that comes naturally from geometry. In fact, we prove these
parametrizations when any base scheme replaces the integers, and show that the
correspondences between forms and the algebraic data are functorial in the base
scheme. We give geometric constructions of the rings and ideals from the forms
that parametrize them and a simple construction of the form from an appropriate
ring and ideal.
