Theorem of Galois

The Fundamental Theorem of Galois connects solutions of polynomial equations to symmetry groups called Galois groups. It states that for certain algebraic equations, there’s a direct relationship between how you can permute the solutions without changing the equation’s structure, and the subsets of solutions that can be found using simpler building blocks called subgroups. This correspondence helps us understand whether equations can be solved using radicals (like square roots), and reveals deep links between algebra, symmetry, and solvability, providing a framework to analyze the solvability of polynomial equations through their symmetries.