The Fundamental Theorem of Algebra via Real Polynomials

Abstract

The fundamental theorem of algebra states that every polynomial p(z) over C of degree m ≥ 1 has a zero in C. There are many proofs of this theorem, but we have not found the elementary one presented here. Splitting the coefficients of p(z) into their real and imaginary parts,we find polynomials a(z) and b(z) with real coefficients such that p(z) = a(z) + ib(z). Then q(z) := p(z)p(z̄) = a(z)2 + b(z)2 is a polynomial of degree 2m ≥ 2 with real coefficients, and q(x) ≥ 0 for all x ∈ R. Moreover, q(z) = 0 if and only if p(z) = 0 or p(z̄) = 0. If q(z) = 0 for all z ∈ C, then F(z) := ∫ 1 0 z/q(tz) dt defines a primitive of 1/q(z) on C. Hence the integral over the piecewise smooth closed curve given by the interval [−r, r] and the positively oriented semi-cirlce Cr := {reiθ : θ ∈ [0, π ]} vanishes, that is, ∫ r −r 1 q(x) dx + ∫