Abstract algebra is a mathematical discipline that studies various abstract axiomatic algebraic systems. Because algebra can handle object sets other than real numbers and complex numbers, such as vectors, matrix supernumbers, transformations, etc. The difference of these object sets depends on their own calculus laws. Mathematicians sublimate some contents of * * * through abstract methods, thus reaching a higher level, thus giving birth to abstract algebra. Abstract algebra includes group theory, ring theory, Galois theory, lattice theory, linear algebra and many other branches. Combined with other branches of mathematics, new mathematical disciplines such as algebraic geometry, algebraic number theory, algebraic topology, topological groups, etc. Abstract algebra has become the universal language of most contemporary mathematics.
Galois (1811-1832) is known as a talented mathematician and one of the founders of modern algebra. He deeply studied the essential condition that an equation can be solved by roots. His Galois Field, Galois Group and Galois Theory are the most important topics in modern algebra research. Galois group theory is recognized as one of the most outstanding mathematical achievements in19th century. He provided a comprehensive and thorough answer to the solvability of equations and solved the problem that puzzled mathematicians for hundreds of years. Galois group theory also gives a general method to judge whether geometric figures can be drawn with straightedge and compass, which satisfactorily solves the problem that bisecting any angle or cube is insoluble. Most importantly, group theory has opened up a brand-new research field, replaced calculation with structural research, changed the way of thinking from emphasizing calculation research to using structural concept research, and classified mathematical operations, which made group theory develop rapidly into a brand-new branch of mathematics and had a great influence on the formation and development of modern algebra. Hamilton invented an algebra-quaternion algebra, in which the multiplicative commutative law does not hold. The following year, grassmann deduced several more general algebras. 1857, Gloria designed another noncommutative algebra-matrix algebra. Their research opened the door to abstract algebra (also called modern algebra). In fact, by weakening or deleting some assumptions of ordinary algebra, or replacing some assumptions with other assumptions (compatible with the rest), we can study various algebraic systems.
In 1870, Kronick gave an abstract definition of finite Abelian groups. Dedekind began to use the term "body" to study algebraic body; 1893, Weber defines abstract body; 19 10 years, Steinitz developed the general abstract theory of the body; Dedekind and Cronic founded the theory of rings; 19 10, Steinitz summarized the research of algebraic systems including groups, algebras and fields, and created abstract algebra.
There is an outstanding female mathematician who is recognized as one of the founders of abstract algebra and is known as the queen of algebra. She is Nott,1born in Hellem, Germany on March 23rd, 882, 1900 entered the University of Herun Gen, 1907 received her doctorate under the guidance of mathematician Gordan.
Nott's work has an important influence on the development of algebraic topology, algebraic number theory and algebraic geometry. In 1907- 19 19, she mainly studies algebraic invariants and differential invariants. In her doctoral thesis, she gave a set of invariants of ternary quartic form. It also solves the existence problem of finite rational basis in rational function domain. The constructive proof that the invariants of finite groups have finite bases is given. She uses direct differentiation instead of elimination to generate differential invariants. In her inaugural thesis at the University of G? ttingen, she discussed the invariants under continuous groups (Lie groups) and gave the Nott theorem, which linked the symmetry, invariance and conservation laws of physics.
From 1920 to 1927, she mainly studied commutative algebra and commutative arithmetic. After 19 16, she began to transition from classical algebra to abstract algebra. In 1920, she introduced the concepts of "left module" and "right module". Written in1921< <; The ideal theory of integral rings is a milestone in the development of commutative algebra. The theory of commutative noetherian rings is established and the quasi-prime decomposition theorem is proved. Published on1926: > An axiomatic characterization of Dai Dejin rings is given, and the necessary and sufficient conditions for the unique decomposition theorem of prime ideal factors are pointed out. Nott's theory is also a systematic theory of "ring" and "ideal" in modern mathematics. It is generally believed that the time of abstract algebra is 1926. Since then, the research object of algebra has changed from studying the calculation and distribution of algebraic equations to studying the algebraic operation rules and various algebraic structures of numbers, words and more general elements, thus completing the essential transformation from classical algebra to abstract algebra. Nott is one of the founders of abstract algebra.
In 1927- 1935, Nott studied noncommutative algebra and noncommutative arithmetic. She unified representation theory, idealism theory and module theory on the basis of so-called "hypercomplex system", that is, algebra. Later, the concept of cross product was introduced and used to determine the brower group of finite dimensional Cangluowa expansion. Finally, the proof of the main theorem of algebra is introduced. The central divisible algebra in algebraic number field is cyclic algebra. It was widely spread through her student Van de Walden's masterpiece & gt. Her main thesis is in>( 1982).
1930, Bierhoff established lattice theory, which originated from Boolean algebra of 1847; After the Second World War, various algebraic system theories and Bourbaki School appeared. 1955, Gadang, Glosindick and Allan Burke established the theory of homology algebra.
Mathematicians have studied more than 200 kinds of such algebraic structures, the most important of which are Jordan algebra and Lie algebra, which are examples of algebras that do not obey the law of association. Most of these works belong to the 20th century, which fully embodies the generalized and abstract ideas in modern mathematics.
Mathematicians in China began to study abstract algebra in 1930s. Significant and important achievements have been made in many aspects, especially the work of Ceng Jiong Zhi, Hua and Zhou Weiliang.