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.