By the18th century, mathematicians could solve equations four times or less. Niels Nott, one of the rare talented female mathematicians in the history of mathematics, strictly proved in 1823 that higher-order equations with more than five times "usually" have no root solution (this point was almost proved to be correct as early as 1799).

However, some quintic equations do have solutions. What is the difference between them and the quintic equation without solution? The French mathematician Galois invented group theory in 1832 and answered this question.

The concept of group captures the essence of symmetry in an abstract form. Every algebraic equation has a symmetric group, namely Galois Group, and its abstract structure determines whether the solution of higher-order equations can be expressed by square roots, cubic roots and other roots. Galois group can tell us which solutions of higher-order equations can be expressed by finite formulas composed of roots, but it cannot give this formula. Nowadays, computer programs can not only calculate the Galois Group of a higher-order equation, but also give the solution formula (if there is a solution).