Cool proof of the Fundamental Theorem of Algebra...

Despite being on holiday I can’t resist looking for cool proofs. This one is not so much cool as interesting in a Why-didn’t-I-think-of-that way. The fundamental theorem of algebra — that any polynomial has a complex root — is well known to be a theorem of analysis rather than algebra and many proofs are known. The proof I use in my course relies on Liouville’s Theorem and the Maximum Modulus Theorem. However, there is a more direct route using only Cauchy’s Integral Formula and the Estimation Lemma. I found the basic idea in Complex Proofs of Real Theorems by Peter Lax and Lawrence Zalcman. Here’s my modified version: Every polynomial , , , has a root in . The proof is as follows. Suppose not and derive a contradiction. Since for all the function defined by is differentiable on all of . Now, for , So there exists an such that implies that . Thus for such . As is differentiable on all of we have for all , by Cauchy’s Integral Formula, (and where denotes a circle of radius centred at the origin), The right hand side can be made as small as we like by taking large enough. Thus . This is impossible so we have a contradiction and therefore has a...