Range of a nonconstant polynomial over complex numbers

Question

If $ P : \mathbb{C} \rightarrow \mathbb{C} $ is a nonconstant polynomail, what is the range of $P$? Prove this.

Wouldn't the range be all of $\mathbb{C}$? If so, I believe I can prove it using contradiction, by assuming the range is $\mathbb{R}$, and then show there exists some $z_0$ such that $P(z_0)$ is not in $\mathbb{R}$

Any help would be great.

Thanks.

Answer 1

Using the fundamental theorem of algebra, we get an easy solution. Let $w\in\mathbb{C}$. Then $P-w$ is a non-constant polynomial, and so has a zero $z_0$ (by the FTA). This means that $P(z_0)=w$, and so $P$ is surjective.