A cubic generalization of Brahmagupta's identity

TL;DR

This paper introduces a new algebraic identity for cubic polynomials that extends Brahmagupta's identity, aiding arithmetic in cubic fields and exploring connections with elliptic curves.

Contribution

It presents a cubic generalization of Brahmagupta's identity and investigates relationships between elements of cubic fields and rational points on elliptic curves.

Findings

New algebraic identity for cubic polynomials

Facilitates arithmetic in cubic fields

Poses questions about cubic field elements and elliptic curves

Abstract

We give an algebraic identity for cubic polynomials which generalizes Brahmagupta's identity and facilitates arithmetic in cubic fields. We also pose a question about a relationship between the elements of a cubic field of fixed trace and fixed norm and rational points of an elliptic curve.