Field Norm of Complex Number is Multiplicative Function

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Let $\C$ denote the set of complex numbers.

Let $N: \C \to \R_{\ge 0}$ denote the field norm on complex numbers:

$\forall z \in \C: \map N z = \cmod z^2$

where $\cmod z$ denotes the complex modulus of $z$.

Then $N$ is a multiplicative function on $\C$.


\(\ds \map N {z_1 z_2}\) \(=\) \(\ds \cmod {z_1 z_2}^2\) Definition of $N$
\(\ds \) \(=\) \(\ds \paren {\cmod {z_1} \cmod {z_2} }^2\) Complex Modulus of Product of Complex Numbers
\(\ds \) \(=\) \(\ds \cmod {z_1}^2 \cmod {z_2}^2\)
\(\ds \) \(=\) \(\ds \map N {z_1} \map N {z_2}\) Definition of $N$

So $N$ is a multiplicative function by definition.