Stickelberger's discriminant theorem for algebras

TL;DR

This paper generalizes Stickelberger's discriminant theorem from number fields to arbitrary finite-rank rings over integers, providing a new elementary proof that broadens the theorem's applicability.

Contribution

It introduces a novel generalization of Stickelberger's discriminant theorem to non-commutative rings using elementary linear algebra techniques.

Findings

Discriminant congruence extends to non-commutative rings

Elementary matrix identities suffice for the proof

The proof is simpler and more general than previous methods

Abstract

Stickelberger proved that the discriminant of a number field is congruent to 0 or 1 modulo 4. We generalize this to an arbitrary (not necessarily commutative) ring of finite rank over the integers using techniques from linear algebra. Our proof, which relies only on elementary matrix identities, is new even in the classical case.