# Visualization of Abel's Impossibility Theorem

## TL;DR

This paper creates an interactive visualization demonstrating Abel's Impossibility Theorem, illustrating why no formula using radicals and analytic functions can solve all polynomial equations of degree five or higher.

## Contribution

It introduces a JavaScript-based visualization tool that graphically illustrates the theorem's proof and implications for polynomial solutions.

## Key findings

- Visualization confirms the theorem's claim for degree five polynomials
- Demonstrates the permutation of roots during analytic continuation
- Shows the necessity of radicals in solving quadratic equations

## Abstract

In this paper we construct a visualization of the Abel's Impossibility Theorem also known as the Abel-Ruffini Theorem. Using the canvas object in JavaScript along with the p5.js library, and given any expression that uses analytic functions and radicals one can always construct closed paths such that the expression evaluated at the coefficients of a general polynomial returns to it's initial position, while the roots of the polynomial undergo a non-trivial permutation. Hence, such expression does not reconstruct the roots from the coefficients. Using the visualization we begin by considering the necessity of radicals to solve second degree polynomial equations and build towards degree five polynomial equations. In eventuality our program shows that there is no formula for an arbitrary fifth degree polynomial equation that uses analytic functions, finite field operations, and radicals that reconstructs the roots of the polynomial from it's coefficients. This theorem was partially completed by Paolo Ruffini in 1799 and completed by Niels Abel in 1824.