Let us suppose that the polynomial

has no root, so that for every complex number z

On this assumption, if we now allow z to describe any closed curve in the x, y - plane, f(z) will describe a closed curve Γ which never passes through the origin.

	Proof of fundamental theorem of algebra.

We may, therefore, define the order of the origin O with respect to the function f(z) for any closed curve C as the net number of complete revolutions made by an arrow joining O to a point on the curve Γ traced out by the point representing f(z) as z traces out the curve C. As the curve C we shall take a circle with O as center and with radius t, and we define the function φ(t) to be the order of O with respect to the function f(z) for the circle about O with radius t. Clearly φ(0) = 0, since a circle with radius 0 is a single point, and the curve Γ reduces to the point f(0) ≠ 0. We shall show in the next paragraph that φ(t) = n for large values of t. But the order φ(t) depends continuously on t, since f(z) is a continuous function of z. Hence we shall have a contradiction, for the function φ(t) can assume only integral values and therefore cannot pass continuously from the value 0 to the value n.

It remains only to show that φ(t) = n for large values of t. We observe that on a circle of radius z = t so large that

we have the inequality

1. R.Courant and H.Robbins, What is Mathematics?, Oxford University Press, 1966.

• Perfect numbers are complex, complex numbers might be perfect
• Fundamental Theorem of Algebra: Statement and Significance
• What's in a proof?
• More about proofs
• Axiomatics
• Intuition and Rigor
• How to Prove Bolzano's Theorem
• Early attempts
• Proofs of the Fundamental Theorem of Algebra
  • Remarks on Proving The Fundamental Theorem of Algebra
    • Sketch of a Proof by Birkhoff and MacLane
    • Sketch of Second Proof (after Cauchy)
    • Details of the Cauchy's Proof
    • Note on the Extreme Value Theorem
    • Sketch of Proof by the methods of the theory of Complex Variables (after Liouville)
    • Maximum Modulus Theorem
  • A Proof of the Fundamental Theorem of Algebra: Standing on the shoulders of giants
  • Yet Another Proof of the Fundamental Theorem of Algebra
  • Fundamental Theorem of Algebra - Yet Another Proof
  • A topological proof, going in circles and counting