Prove that $f(z)=z^4$ is a group homomorphism

Question

If $f:(\mathbb C^*,.) \to (\mathbb C^*,.)$ is defined by

$f(z)=z^4$

1- Prove that $f$ is a group homomorphism.

2- find $\mathrm{ker}(f)$.

1- $f(z.y)=(zy)^4=z^4.y^4$

$f(z)=z^4$, $f(y)=y^4$

$f(z).f(y)=z^4.y^4$

So $f(z.y)=f(z).f(y)$

Then $f$ is a group homomorphism.

True ?

And what about $\mathrm{ker}(f)$ ?

Answer 1

The first part is good, you have shown that $f$ preserves products. To calculate the kernel, $ker=\{z | z^4=1\}$. These are $\{1,-1,i,-i\}$, check by definition.