Fundamental theorem of algebra

Definition

Let $p:\mathbb{C}\to \mathbb{C}$ be a polynomial function; in other words, there exist $a_0,a_1,\dots ,a_n\in \mathbb{C}$ 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.