A Closed Formula for the Product in Simple Integral Extensions

TL;DR

This paper derives an explicit closed-form formula for multiplying elements in simple integral extensions, expressed via coordinates and the companion matrix of the minimal polynomial, applicable to broad classes of such extensions.

Contribution

It provides a new explicit formula for the product in simple integral extensions, extending previous results to more general cases.

Findings

Explicit formula for product coordinates in simple integral extensions

Formula expressed via minimal polynomial's companion matrix

Applicable to broad classes of algebraic extensions

Abstract

Let $\xi$ be an algebraic number and let $\alpha,\beta\in \mathbb Q[\xi]$. An explicit formula for the coordinates of the product $\alpha\beta$ is given in terms of the coordinates of $\alpha$ and $\beta$ and the companion matrix of the minimal polynomial of $\xi$. The formula as well as its proof extend to fairly general simple integral extensions.