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}

Read more about this topic:  Eisenstein Integer

Famous quotes containing the word domain:

    “When it had long since outgrown his purely medical implications and become a world movement which penetrated into every field of science and every domain of the intellect: literature, the history of art, religion and prehistory; mythology, folklore, pedagogy, and what not.”
    —Thomas Mann (1875–1955)