Continuity of the roots of a polynomial

TL;DR

This paper provides an elementary proof demonstrating that the roots of a polynomial vary continuously with its coefficients over an algebraically closed field with an absolute value.

Contribution

It offers a straightforward proof of the classical root continuity theorem, simplifying understanding of how polynomial roots depend on coefficients.

Findings

Roots depend continuously on coefficients in algebraically closed fields with absolute value

Elementary proof simplifies classical understanding

Reinforces the stability of roots under coefficient perturbations

Abstract

Let $K$ be an algebraically closed field with an absolute value. This note gives an elementary proof of the classical result that the roots of a polynomial with coefficients in $K$ are continuous functions of the coefficients of the polynomial.