UWEC CERCA 2026

This project examines why general polynomial equations of degree five cannot be solved using radicals. The aim of this research is to explain the algebraic structures that prevent the existence of a general radical formula for the quintic equation.The problem of solving polynomial equations has a long mathematical history. While formulas for quadratic, cubic, and quartic equations were discovered, mathematicians later proved that no similar formula exists for the general quintic. This result was first established by Niels Henrik Abel and later explained structurally through Galois Theory developed by Évariste Galois. Understanding this shift from formula-based algebra to structural reasoning is central to modern abstract algebra.The presentation introduces key ideas from field extensions and permutation groups and explains how the solvability of a polynomial relates to the structure of its Galois group. In particular, the project discusses why the general quintic has a Galois group isomorphic to S_5, which is not solvable.This work aims to clarify the algebraic reasons behind the insolvability of the quintic and highlight the significance of Galois Theory in understanding polynomial equations.