Please provide me the proof of fundamental theorem of Algebra.
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Answers
♦ The Fundamental Theorem of Algebra is the statement
Every non constant polynomial has at least one zero in CC .
The most commonly found proof of this uses Liouvi-lle’s theorem:
Every bounded entire function is a constant.
The proof I am going to give can be found in any standard textbook on Complex Analysis.
Let, p (z) € C (z) , with deg f = n > 0 and suppose
p (z) ≠ 0 for all z € C. then,
f (z) = 1 / p(z).
is a entire, since it is a quotient of two analytic functions with a denominator never equal to 0 .
.The function f(z) is also bounded on C . To see this, note that as z→∞ , ∣p(z)∣ →∞ | p(z) | →∞ and so ∣ f(z)∣ →0 | f(z) |→0 . Consequently, there exists Rϵ ∈ R ϵ∈ R for which
R for which∣ f (z) ∣ <ϵ |f ( z) | < ϵ whenever |z| >Rϵ | z| >Rϵ .
I don't know, you can take help from Goo-gle.
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