The Fundamental Theorem of Algebra

The fact that polynomials are complex analytic in the entire complex plane lies at the heart of a beautiful and simple proof of the fundamental theorem of algebra.

Although it might appear like we are hiding the complexity of the proof by building up so much machinery that it's possible dispose of this mighty theorem in a few lines, in fact it is truly direct. For the most powerful ideas are necessarily the simplest; if they were not simple it would not be possible for them to explain so much.

[1] Complex Analysis, Elias M. Stein, Rami Shakarchi (Chapters 1 & 2)
https://books.google.com/books?id=0EC...

[2] A Course of Modern Analysis, E. T. Whittaker, G. N. Watson (Chapter 5)
https://books.google.com/books?id=ULV...

[3] Geometry and the Imagination, David Hilbert, Stephan Cohn-Vossen (Chapter 4, section 38)
https://books.google.com/books?id=7WY...