= Eisenstein norm
{c}
{title2=$N(a+b\omega)=a^2-ab+b^2$}
= Eisenstein-integer norm
{c}
{synonym}
For an <Eisenstein integer> $z=a+b\omega$, with $\omega=(-1+i\sqrt3)/2$, the norm is $N(z)=z\overline z=a^2-ab+b^2$. It is a positive <integer> for $z\ne0$, since $N(z)=(a-b/2)^2+3b^2/4$, and is multiplicative by the multiplicativity of the <complex modulus>. An element is a <unit> exactly when its norm is one: one direction follows from multiplicativity and integrality, and the other from the inverse $z^{-1}=\overline z$.
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