Fundamental theorem of algebra

Definition

Let $p : C \to C$ be a polynomial function; in other words, there exist $a_{0}, a_{1}, \dots, a_{n} \in C$ such that for all $z \in C$ , we have:

$p (z) : = \sum_{r = 0}^{n} a_{r} z^{r}$

Then, there exists a $z \in C$ such that $p (z) = 0$ .

Bounded and entire implies constant: Any bounded entire function (function holomorphic on all the complex numbers) must be constant.