Motivated by Zorn Leung’s post, let me show another proof of the famous Fundamental theorem of algebra. It is also mentioned in a mathoverflow post. The proof can be found in a paper by Terkelsen (1976), but the idea should have appeared in earlier works.

**Theorem**** (Fundamental theorem of algebra)**

*Every polynomial with leading coefficient has a complex root. *

*Proof.* Consider minimizing the function given by

.

We have for ,

As , the sum above converges to . Thus . By this and elementary analysis, one sees that has a minimizer .

First we assume . Then as is a minimizer of ,

.

Hence is the minimum value of . If , then is a root of and we are done. If , then we write

where is the first nonzero coefficient after and is a polynomial. Let be a k-th root of . Then for any ,

.

If is small, then is small. Hence for a sufficiently small , , and so

which is a contradiction. We have justified if is the minimizer of , then .

If , then consider the polynomial . One has that is a minimizer of . By the above case, and hence . The proof is completed.

Very nice! In my view this is the best proof of FTA in that nearly no knowledge of complex analysis is required, except that we can take the k-th root of any number.

Agree. I think the proof (or the idea of the proof) can be taught to some “smart” high school students.

Very nice!

Very nice

Small typo: f(z) should be instead of . And in the statement of FTA, p should be non-constant (of course).

Could you show formally why the sum converges to a_n and why f(z) has a minimizer z^* ? I really like your proof, but I need to understand these two steps.

As , if , thus the finite sum converges to .

As as , there is a sufficiently large closed ball centered at such that is large (say ) outside . As is closed and bounded, a minimizer of in exists and is actually a minimum point of .

Could you point me to some theorem/lemma proving that f has a minimizer in B if it is closed and bounded ?

Also, could one show formally that the sum converges to |a_n| using epsilon without to many trouble ?

For the first question, see here:

http://en.wikipedia.org/wiki/Extreme_value_theorem

or more precisely here:

http://en.wikipedia.org/wiki/Compact_space#Theorems

See also the Heine-Borel theorem.

For any constant and , if , then . Using this, and the triangle inequality, I think you should have no trouble working out the estimate.

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