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# Norm

The norm ${\displaystyle N(z)}$ of an algebraic integer ${\displaystyle z=a+b{\sqrt {d}}}$ is ${\displaystyle a^{2}-db^{2}}$. For example, the norm of ${\displaystyle 5+2{\sqrt {7}}}$ is –3. If ${\displaystyle N(z)=\pm 1}$, then ${\displaystyle z}$ is one of the units in ${\displaystyle \mathbb {Z} [{\sqrt {d}}]}$.
If ${\displaystyle d<0}$, then ${\displaystyle N(z)=a^{2}+|d|b^{2}}$, meaning that the norm of an integer in a quadratic imaginary ring is never negative. For example, ${\displaystyle N(3+2{\sqrt {-5}})=3^{2}-|-5|2^{2}=3^{2}+5\times 2^{2}=29}$.