In $\mathbb{C}$, prove that $z_1z_2z_3=0 \Rightarrow z_1=0$ or $z_2=0$ or $z_3=0$

Question

I could prove this but I'd like to see other ways.

My prove was something like (by contrapositive):

Suppose they are all non-zero elements of $\mathbb{C}$, hence they have their inverses. And now, suppose that could be the case that $z_1z_2z_3=0$, but then $ 0=z_2^{-1}z_1^{-1}z_1z_2z_3=z_3 $, contradiction.

I am asking myself if this could be provedd by using the definition of multiplication in $\mathbb{C}$, so I'd like to see other proofs for this.

What are your ideas?

Answer 1

For any $z\in\mathbb{C}$ we have $z=0$ if and only if $|z|=0$. Therefore $$z_1z_2z_3=0\Leftrightarrow |z_1z_2z_3|=0\Leftrightarrow |z_1||z_2||z_3|=0\Rightarrow |z_1|=0\hspace{0.1cm}\text{or}\hspace{0.1cm}|z_2|=0\hspace{0.1cm}\text{or}\hspace{0.1cm}|z_3|=0$$ if and only if $z_1=0$ or $z_2=0$ or $z_3=0$.

Answer 2

In any field $K$, $z_1\cdots z_n=0$ implies that one of the $z_i$ is zero, because a field has no nontrivial zero divisors. So this does not depend on the complex numbers, but holds in general.