Galois theory is a theory that relates the algebraic structures of group and field, and it is one of the basic theories of algebra and even modern mathematics. The establishment of this theory is a milestone in the development of algebra, which brings great innovation in the research object, content and method of algebra.

Galois theory originated from solving algebraic equations, which was the core problem of algebraic research before19th century. The roots of algebraic equations are solvable, that is, the solutions of the equations can be expressed by finite addition, subtraction, multiplication, division and integer power operations of the coefficients of the equations.

As early as in ancient Greece, the root formula of quadratic equation was obtained. By the middle of16th century, a group of Italian mathematicians had given the root solutions of cubic equation and quartic equation successively. In the next two hundred years, mathematicians devoted themselves to exploring the root solutions of quintic and above equations.

It was not until 1826 that Abel strictly proved that algebraic equations greater than quartic had no radical solution. The French mathematician Galois, who was almost contemporary with Abel, gave the necessary and sufficient conditions for the root solution of the algebraic equation around 1830, and solved this problem. The theoretical method developed from this was called Galois Theory by later generations.

This theory can lead to the conclusion that the roots of general algebraic equations with more than five degrees are insoluble, and it is impossible to bisect any angle with compasses and straightedge (scale-free ruler) and make a cube. Galois theory is a theory that relates the algebraic structures of group and field, and it is one of the basic theories of algebra and even modern mathematics.

The establishment of this theory is a milestone in the development of algebra, which brings great innovation in the research object, content and method of algebra. Galois published his first paper on continued fractions. Galois put forward the concept of group, thoroughly solved the problem of solving algebraic equations with roots by group theory, and developed a set of theories about groups and fields.

In memory of him, later generations called this theory Galois Theory. Galois revised the manuscript of 183 1, added some marginal notes, and entrusted the manuscript to his friend Chevalier.