Galois groups were developed to help solve polynomial equations by capturing their symmetry features. For example, the solutions of the quadratic equation a x 2 + b x + c = 0 are given by x = − − b ± ± b 2 − − 4 a c 2 a . Each solution can be obtained by replacing the ± ± sign by + or − − ; analogous formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. In the quadratic formula, changing the sign (permuting the resulting two solutions) can be viewed as a (very simple) group operation. Analogous Galois groups act on the solutions of higher-degree polynomial equations and are closely related to the existence of formulas for their solution. Abstract properties of these groups (in particular their solvability) give a criterion for the ability to express the solutions of these polynomials using solely addition, multiplication, and roots similar to the formula above.