In the second half of the18th century, faced with a series of long-standing unresolved problems in the development of mathematics, such as solving the roots of algebraic equations higher than quartic, intense discussions and concerns were launched, which led to the unprecedented rapid development of mathematics in the19th century.
Here is a brief introduction to the solvability of algebraic equations and the discovery of groups:
Mathematicians in the Middle Ages regarded algebra as the knowledge of solving algebraic equations. Until the beginning of19th century, the study of algebra was still not beyond this range, but mathematicians focused on algebraic equations of quintic and higher order. After the algebraic equation has a degree higher than four, can it be obtained just by adding, subtracting, multiplying and dividing the square root of a positive integer, like the equation of 234? For two and a half centuries, few people doubted the possibility of solving equations with roots of five or more.
The first mathematician who explicitly declared that "it is impossible to solve equations with more than four degrees" was Lagrange. He discussed all the well-known solutions of quadratic, cubic and quartic equations in the article "Reflections on the Solutions of Algebraic Equations" published by 1770, and then announced that the conditions on which these solutions are based are not applicable to equations of quintic and above. But he didn't prove this impossibility.