Isomorphisms of Discriminant Algebras

TL;DR

This paper investigates the relationships between various definitions of discriminant algebras across different ranks, establishing their isomorphisms and uniqueness in low ranks, thus unifying existing concepts.

Contribution

It introduces a categorical framework for discriminant algebras and proves their isomorphism and uniqueness in ranks up to three, unifying prior definitions.

Findings

Discriminant algebras in different definitions are isomorphic within the defined category.

Discriminant algebras are unique up to isomorphism for ranks n ≤ 3.

The paper provides a categorical perspective on discriminant algebra equivalences.

Abstract

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 $n \leq 3$ discriminant algebras are unique up to unique isomorphism.