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# 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 root.