proof of fundamental theorem of algebra (due to Cauchy)
We will prove that any equation
Proof. We can suppose that . Denote where are real. Then the function
Hence is the absolute minimum of in the whole complex plane. We show that . Therefore we make the antithesis that .
Denote , and
Then by the antithesis. Moreover, denote
and assume that but Thus we may write
If and , then
by de Moivre identity. Choosing and we get and can make the estimation
where is a constant. Let now . We obtain
which result is impossible since was the absolute minimum. Consequently, the antithesis is wrong, and the proof is settled.
|Title||proof of fundamental theorem of algebra (due to Cauchy)|
|Date of creation||2013-03-22 19:11:10|
|Last modified on||2013-03-22 19:11:10|
|Last modified by||pahio (2872)|
|Synonym||Cauchy proof of fundamental theorem of algebra|