Abstract algebra has an important influence on all modern mathematics and other scientific fields. Abstract algebra is developing with the development and application of various branch theories in mathematics. Following the work of boekhoff, von Neumann, kantorovich and Si Tong in 1933- 1938, lattice theory established its position in algebra. Since the mid-1940s, as a generalization of linear algebra, modular theory has been further developed and exerted a far-reaching influence. New fields such as universal algebra, homology algebra and category have also been established and developed. Abstract algebra has made a good start in the last century, and Galois has already included the concept of group when finding the roots of equations. Later, Kelly gave an abstract definition of group (Cayley, 182 1~ 1895). He put forward the concept of abstract group in a work of 1849, but unfortunately it did not arouse repercussions. Premature abstraction entered the deaf ear. It was not until 1878 that Kelly wrote four more articles about abstract groups that attracted attention. 1874, Norwegian mathematician sophoslie (1842 ~1899) found that some solutions of differential equations are invariant to some continuous transformation groups, and they came into contact with continuous groups at once. 1882, von dyke of Britain (1856 ~ 1934) unified the three main sources of group theory-equation theory, number theory and infinite transformation group-into one concept and put forward the concept of "generator". At the beginning of the 20th century, the abstract axiom system of groups was given.

The study of group theory was carried out from different directions in the 20th century. For example, find all finite groups of a given order. The decomposition of groups into simple groups and solvable groups is studied. The classification of finite simple groups was obtained in 1970s and 1980s, which may be the ultimate solution. Burnside (1852~ 1927) raised many questions and conjectures. For example, in 1902, the group G is generated finitely, and each element is finite. Is g a finite group? And guess that every noncommutative simple group is even order. The former has not been solved, and the latter was solved in 1963.

Schur (1875 ~1941) put forward the problem of finite group representation in 190 1. The study of group characteristics was first put forward by Frobenius. Poincare has a special passion for group theory. He said, "group theory is the whole mathematics that abandons its content and turns it into a pure form." This is of course an exaggeration.

Another part of abstract algebra is field theory. 19 10, Steinitz published Algebraic Theory of Fields, which became an important milestone in abstract algebra. He put forward the concept of prime domain, defined the domain with the characteristic number p, and proved that each domain can be obtained by adding its prime domains.

Ring theory is a late mature theory in abstract algebra. Although rings and ideal structures can be found in19th century, abstract theory is the product of 20th century. Wedderburn (1882~ 1948) studied the linear associative algebra in On Hypercomplex Numbers, which is actually a ring. Nott gave the theory of rings and ideal systems. When she began to work, many results of rings and ideals already existed, but when she gave them proper and accurate expressions, she got abstract theories. Nott included the ideal theory of polynomial rings in the general ideal theory, which laid the same foundation for the ideal theory of algebraic integers and the ideal theory of algebraic whole functions. Nott made a profound study of rings and ideals. It is considered that this summative work was completed in 1926, so abstract algebra was formed in 1926. Vander Waals Deng wrote Modern Algebra according to the handouts of Nott and Anting (1955 was renamed Algebra in the fourth edition), and his research object shifted from studying the calculation and distribution of algebraic equation roots to studying algebraic operation rules and various algebraic structures of numbers, words and more general elements. This has undergone a qualitative change. Because of the generality of abstract algebra, its methods and results have basic properties, so it permeates different branches of mathematics. Deng's Algebra is still a good book for studying algebra. People have drawn nutrition from the brilliant achievements of the founders of abstract algebra, such as Nott and Anting. Since then, great progress has been made in algebra research.