Field Norm of Complex Number is Multiplicative Function

Theorem

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 $\C$.

Proof

\(\ds \map N {z_1 z_2}\)

\(=\)

\(\ds \cmod {z_1 z_2}^2\)

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.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $9$: Rings: Exercise $19$