Isomorphisms of discriminant algebras

For each integer n >= 0, we define a category whose objects are discriminant algebra functors in rank n, namely, choices of how to attach functorially to each rank-n algebra a quadratic algebra with the same discriminant. We show that the discriminant algebra functors defined by Loos, Rost, and the present authors are all isomorphic in this category, and prove furthermore that in ranks n <= 3 discriminant algebra functors are unique up to unique isomorphism.