Fundamental theorem of algebra

Definition

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

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

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

Facts used

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