Eisenstein Integer - Euclidean Domain

Euclidean Domain

The ring of Eisenstein integers forms a Euclidean domain whose norm N is given by

This can be derived as follows:

\begin{align}N(a+b\,\omega) &=|a+b\,\omega|^2\\ &=(a+b\,\omega)(a+b\,\bar\omega)\\ &=a^2 + ab(\omega+\bar\omega) + b^2\\ &=a^2 - ab + b^2\end{align}

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