proof of fundamental theorem of algebra (due to Cauchy)

We will prove that any equation

f(z):=zn+a1zn-1+a2zn-2++an-1z+an= 0,

where the coefficients aj are complex numbersMathworldPlanetmathPlanetmath and  n1,  has at least one root ( in .

Proof.  We can suppose that  an0.  Denote  z:=x+iy  where  x,y are real.  Then the function


is defined and continuousMathworldPlanetmath in the whole 2.  Let  c:=j=1n|aj|; it is positive.  Using the triangle inequalityMathworldMathworldPlanetmath we make the estimation

|f(z)| =|z|n|1+a1z+a2z2++anzn|

being true for  |z|>max{1, 2c}.  Denote  r:=max{1, 2c,2|an|n}.  Consider the disk  x2+y2r2.  Because it is compact, the function  g(x,y)  attains at a point  (x0,y0)  of the disk its absolute minimum value (infimum) in the disk.  If  |z|>r,  we have

g(x,y)=|f(z)|12|z|n>12rn12(2|an|n)n=|an|> 0.


g(x0,y0)g(0, 0)=|an|<|f(z)|for|z|>r.

Hence g(x0,y0) is the absolute minimum of g(x,y) in the whole complex planeMathworldPlanetmath.  We show that  g(x0,y0)=0.  Therefore we make the antithesis that  g(x0,y0)>0.

Denote  z0:=x0+iy0,   z:=z0+u  and


Then  bn=f(z0)0  by the antithesis.  Moreover, denote

cj:=bjbn(j= 1, 2,,n),c0:=1bn.

and assume that  cn-1=cn-2==cn-k+1=0  but  cn-k0.  Thus we may write


If  cn-k=p(cosα+isinα)  and  u=ϱ(cosφ+isinφ), then


by de Moivre identityMathworldPlanetmath.  Choosing  ϱ1  and  φ=π-αk  we get  cn-kuk=-pϱk  and can make the estimation


where R is a constant.  Let now  ϱ=min{1,1pk,p2R}.  We obtain

|f(z)| =|bn||1-pϱk+h(u)|

which result is impossible since |f(z0| was the absolute minimum.  Consequently, the antithesis is wrong, and the proof is settled.

Title proof of fundamental theorem of algebra (due to Cauchy)
Canonical name ProofOfFundamentalTheoremOfAlgebradueToCauchy
Date of creation 2013-03-22 19:11:10
Last modified on 2013-03-22 19:11:10
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Proof
Classification msc 30A99
Classification msc 12D99
Synonym Cauchy proof of fundamental theorem of algebra