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1.4 The fundamental theorem of algebra

Given complex numbers $a_{0},\dots,a_{n-1}$ , the equation

z^{n}+a_{n-1}z^{n-1}+\dots+a_{1}z+a_{0}=0

has a complex root.

The proof will be in MATH215 Complex Analysis.

There exist $\alpha_{1},\dots,\alpha_{n}\in{\mathbb{C}}$ , not necessarily all distinct, such that

z^{n}+a_{n-1}z^{n-1}+\dots+a_{1}z+a_{0}=(z-\alpha_{1})(z-\alpha_{2})\dots(z-% \alpha_{n}).

Proof. By induction on $n$ .