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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
Thus
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