A Closed Formula for the Product in Simple Integral Extensions

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

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