Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

Let z0 be a zero of g(z). Since g(z) is analytic at z0, we can expand g(z) in a power series around z0:

g(z) = Σn=0∞ a_n(z - z0)^n

where the coefficients a_n are given by:

a_n = 1/n! * g^(n)(z0)

where g^(n)(z0) denotes the nth derivative of g(z) evaluated at z0.

Since g(z) has a zero at z0, we know that a_0 = 0. Now, we can define a new function h(z) as follows:

h(z) = a_1 + a_2(z - z0) + a_3(z - z0)^2 + ...

This is a polynomial function of degree at most n - 1, where n is the degree of f(z). Furthermore, since g(z) has a zero at z0, we can see that h(z0) = a_1 ≠ 0.

Now, we can consider the behavior of h(z) as |z| approaches infinity. Since g(z) has a pole at infinity, we know that the coefficients a_n must approach zero as n → ∞. Therefore, h(z) must approach a constant as |z| → ∞.

By the fundamental theorem of algebra (which we will prove shortly), we know that h(z) has at least one complex root, say

Since the complex plane is a simply connected domain, we can now construct a closed curve γ that encloses all of the disks around the roots z1, z2, ..., zk. We can then apply the argument principle from complex analysis to count the number of zeros of g(z) inside the curve γ.

The argument principle states that the number of zeros of g(z) inside γ is equal to the change in the argument of g(z) as z moves around the curve γ. Since |g(z)| > 1 for all z inside γ, we know that the argument of g(z) changes by an integer multiple of 2π as z moves around γ.

Therefore, the number of zeros of g(z) inside γ is an integer multiple of 2π divided by 2πi, which is an integer. But we have already shown that g(z) has at least one zero in the complex plane, which means that the number of zeros inside γ must be positive. This is a contradiction, which means that our assumption that f(z) has no complex roots must be false.

Therefore, every non-constant polynomial with complex coefficients has at least one complex root. This is the Fundamental Theorem of Algebra, and our proof combines ideas from complex analysis, topology, and algebra to provide a unique and comprehensive approach.