information:extraction
Galois groups

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 {\displaystyle ax^{2}+bx+c=0} are given by x = − b ± b 2 − 4 a c 2 a. {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.} Each solution can be obtained by replacing the ± {\displaystyle \pm } sign by + {\displaystyle +} or − {\displaystyle -}; 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.

Modern Galois theory generalizes the above type of Galois groups by shifting to field theory and considering field extensions formed as the splitting field of a polynomial. This theory establishes—via the fundamental theorem of Galois theory —a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.