A short proof of d'Alemberts theorem

TL;DR

This paper presents a concise, elementary inductive proof of d'Alembert's theorem, demonstrating that polynomials with real coefficients having a non-real root also have its conjugate, without relying on complex conjugation properties.

Contribution

The paper introduces a novel, simplified inductive proof of d'Alembert's theorem that does not depend on complex conjugation properties.

Findings

Proof is shorter and more elementary than traditional methods

Avoids reliance on properties of complex conjugation

Provides a new perspective on classical polynomial root symmetry

Abstract

A classical theorem of d'Alembert states that if a polynomial P(x) with real coefficients has a non-real root x=a+ib, then it also has a root x=a-ib. We give a short and elementary inductive proof that avoids any properties of the complex conjugation operator.