The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is on of the most important theorems in analysis. It states:

Ifforwiththen has at least one zero. It can be used to prove thathaszeros quite easily.

Proof: Suppose thatfor allThenis entire It is also bounded since

Any bounded entire function is constant by Liouville's Theorem and sois constant. This is a contradition sinceis a polynomial of degree greater than 1 so must have at least one zero.

Suppose is this zero, then we can write

We can apply the argument repeatedly toto prove that a polynomial of orderhaszeros in

It is important to not that the roots are in general complex. In fact a polynomial may not have any real roots at all. For example ifthen the roots areandThe fundamental theorem only guarantees complex roots, not real roots.

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