# Fundamental Theorem of Algebra.

As a warm-up, I want to give probably the most beautiful proof of the Fundamental Theorem of Algebra which I know, using the theory of one complex variable. In case you don't know the theorem:

Theorem (Fundamental Theorem of Algebra).

Let be a polynomial, . If , there exist constants , such that

The main ingredient of the proof is the following statement, which is in fact eqiuvalent to the Fundamental Theorem:

Lemma.

Let be a non-constant polynomial. Then there exists an with .

Proof.

Assume that on the contrary, is zero-free. In that case, defines a entire function, i.e. a function defined on which is holomorphic everywhere. We will show that is bounded, whence it follows by Liouville's Theorem that is constant. This implies that is constant, a contradiction.

First, write with ; then

Clearly, for , uniformly, whence uniformly for . Therefore, is bounded on for some .

Now consider on . We have that is continuous on , and since is compact, we know that attains its maximum on , whence is bounded on .

Therefore, is bounded on , and we can conclude.

Now we are able to prove the Fundamental Theorem:

Proof (Fundamental Theorem).

We proceed by induction on , the degree of . If , then , whence we can set and .

Now assume that the statement holds for polynomials of degree . Let be a polynomial of degree . By the lemma, there exists some with . Now, using the Division Algorithm, write with polynomials satisfying , i.e. . Now

whence we have and, therefore, . As , we have .

Therefore, by the induction hypothesis, there exist , with , whence

i.e. the induction hypothesis holds for , too.