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 functions. 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) (http://planetmath.org/ProofOfFundamentalTheoremOfAlgebra3).
Let denote the degree of . Without loss of generality, the assumption can be made that the leading coefficient of is . Thus, .
Let . Note that, by choice of , whenever , . Suppose that . Since , whenever . Hence, we have the following of inequalities:
Since polynomials in are entire, they are certainly analytic functions in the disk . Thus, Rouché’s theorem can be applied to them. Since for , Rouché’s theorem yields that and have the same number of zeroes in the disk . Since has a single zero of multiplicity at , which counts as zeroes, must also have zeroes counted according to multiplicity in the disk . By choice of , it follows that has exactly zeroes in the complex plane. ∎
|Title||proof of fundamental theorem of algebra (Rouché’s theorem)|
|Date of creation||2013-03-22 14:36:21|
|Last modified on||2013-03-22 14:36:21|
|Last modified by||Wkbj79 (1863)|