A discriminant algebra operation sends a commutative ring R and an R-algebra A of rank n to an R-algebra ∆ A/R of rank 2 with the same discriminant bilinear form. Constructions of discriminant algebra operations have been put forward by Rost, Deligne, and Loos. We present a simpler and more explicit construction that does not break down into cases based on the parity of n. We then prove properties of this construction, and compute some examples explicitly.

Let R be a ring and let A be a finite projective R-algebra of rank n. Manjul Bhargava and Matthew Satriano have recently constructed an R-algebra G(A/R), the Galois closure of A/R. Many natural questions were asked at the end of their paper. Here we address one of these questions, proving the existence of the natural constructions they call intermediate Sn-closures. We will also study properties of these constructions, generalizing some of their results, and proving new results both on the intermediate Sn-closures and on G(A/R).

A discriminant algebra operation sends a commutative ring R and an R-algebra A of rank n to an R-algebra ∆ A/R of rank 2 with the same discriminant bilinear form. Constructions of discriminant algebra operations have been put forward by Rost, Deligne, and Loos. We present a simpler and more explicit construction that does not break down into cases based on the parity of n. We then prove properties of this construction, and compute some examples explicitly.

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For each natural number n, we define a category whose objects are discriminant algebras in rank n, i.e. functorial means of attaching to each rank-n algebra a quadratic algebra with the same discriminant. We show that the discriminant algebras defined in [2], [6], and [10] are all isomorphic in this category, and prove furthermore that in ranks n ≤ 3 discriminant algebras are unique up to unique isomorphism.