Norm

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

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, $N(3+2\sqrt{-5})=3^2-|-5|2^2=3^2+5\times 2^2=29$.