A Closed Formula for the Product in Simple Integral Extensions

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.