Norm

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The norm $N(z)$ of an algebraic integer $z=a+b{\sqrt {d}}$ is $a^{2}-db^{2}$. For example, the norm of $5+2{\sqrt {7}}$ is –3. If $N(z)=\pm 1$, then $z$ is one of the units in $\mathbb {Z} [{\sqrt {d}}]$.

If $d<0$, then $N(z)=a^{2}+|d|b^{2}$, meaning that the norm of an integer in a quadratic imaginary ring is never negative. For example, $N(3+2{\sqrt {-5}})=3^{2}-|-5|2^{2}=3^{2}+5\times 2^{2}=29$.