Algebraic norm ($N(\alpha )$)

The algebraic norm of an element of a finite field extension is the determinant of multiplication by that element. In a quadratic subring of $\mathbb{C}$, it is often $N(\alpha )=\alpha \overline{\alpha }$.

For algebraic elements $\alpha$ and $\beta$,This follows because multiplication by $\alpha \beta$ is the composition of multiplication by $\beta$ and multiplication by $\alpha$, and determinants are multiplicative.