Galois Groups

Galois groups are mathematical structures that describe how the solutions of polynomial equations relate to each other through symmetries. They identify the ways you can rearrange or permute the roots (solutions) without changing the overall structure of the equation. These groups help mathematicians understand whether solutions can be expressed using basic operations (like square roots) and play a key role in solving problems about the solvability of equations. In essence, Galois groups connect the algebraic properties of equations with symmetry patterns, revealing deep insights into the nature and solvability of polynomial equations.