A One-Line Proof of the Fundamental Theorem of Algebra with Newton's Method as a Consequence

TL;DR

This paper presents a concise proof of the fundamental theorem of algebra by demonstrating a descent direction for polynomial modulus, linking it directly to Newton's method for solving polynomial equations.

Contribution

It introduces a novel, simplified proof of the theorem that naturally incorporates Newton's method as a consequence.

Findings

A simple proof of the existence of descent directions for polynomial modulus.

Connection established between descent directions and Newton's method.

Iterates are well-defined at critical points in the proof.

Abstract

Many proofs of the fundamental theorem of algebra rely on the fact that the minimum of the modulus of a complex polynomial over the complex plane is attained at some complex number. The proof then follows by arguing the minimum value is zero. This can be done by proving that at any complex number that is not a zero of the polynomial we can exhibit a direction of descent for the modulus. In this note we present a very short and simple proof of the existence of such descent direction. In particular, our descent direction gives rise to Newton's method for solving a polynomial equation via modulus minimization and also makes the iterates definable at any critical point.