proof of fundamental theorem of algebra (Rouché’s theorem)

The fundamental theorem of algebra can be proven using Rouché’s theorem. Not only is this proof interesting because it demonstrates an important result, it also serves to provide an example of how to use Rouché’s theorem. Since it is quite , it can be thought of as a “toy model” (see toy theorem) for theorems on the zeroes of analytic functionsMathworldPlanetmath. For a variant of this proof in of the argument principle (of which Rouché’s theorem is a consequence), please see the proof of the fundamental theorem of algebra (argument principle) (


Let n denote the degree of f. Without loss of generality, the assumption can be made that the leading coefficient of f is 1. Thus, f(z)=zn+m=0n-1cmzm.

Let R=1+m=0n-1|cm|. Note that, by choice of R, whenever |z|>R, f(z)0. Suppose that |z|R. Since R1, |za||zb| whenever 0<a<b. Hence, we have the following of inequalitiesMathworldPlanetmath:


Since polynomials in z are entire, they are certainly analytic functions in the disk |z|R. Thus, Rouché’s theorem can be applied to them. Since |m=0n-1cmzm||zn| for |z|R, Rouché’s theorem yields that zn and f(z) have the same number of zeroes in the disk |z|R. Since zn has a single zero of multiplicity n at z=0, which counts as n zeroes, f(z) must also have n zeroes counted according to multiplicity in the disk |z|R. By choice of R, it follows that f has exactly n zeroes in the complex planeMathworldPlanetmath. ∎

Title proof of fundamental theorem of algebra (Rouché’s theorem)
Canonical name ProofOfFundamentalTheoremOfAlgebraRouchesTheorem
Date of creation 2013-03-22 14:36:21
Last modified on 2013-03-22 14:36:21
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 14
Author Wkbj79 (1863)
Entry type Proof
Classification msc 30A99
Classification msc 12D99