A short proof of the Fundamental Theorem of Algebra

TL;DR

This paper presents a shorter, clearer elementary proof of the Fundamental Theorem of Algebra, building on D'Alembert's original idea and improving upon previous proofs for better transparency.

Contribution

It provides a more concise and transparent elementary proof of the FTA, enhancing understanding and accessibility of the theorem.

Findings

A shorter, more transparent proof of the FTA is constructed.

The proof simplifies previous approaches, making the theorem more accessible.

The approach is based on ideas similar to those in 'Proofs from THE BOOK'.

Abstract

D'Alembert made the first serious attempt to prove the Fundamental Theorem of Algebra (FTA) in 1746. An elementary proof of (FTA) based on the same idea is given in Proofs from THE BOOK. We give a shorter and more transperant version of this proof.