The association of algebraic objects to forms has had many important applications in number theory. Gauss, over two centuries ago, studied composition of binary quadratic forms, which we now understand via Dedekind's association of ideal classes of quadratic rings to integral binary quadratic forms. Delone and Faddeev, in 1940, showed that cubic rings are parameterized by equivalence classes of integral binary cubic forms. Birch, Merriman, Nakagawa, del 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 parameterized by binary n-ic forms, for all n. The algebraic data associated to an integral binary n-ic form includes a ring isomorphic to _n as a -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 parameterizations 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 parameterize them and a simple construction of the form from an appropriate ring and ideal.
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