The Evolution Of The Idea Of Finite Fields Theory

As a branch of modern algebra , finite fields is an abstract system in algebra which satisfies operational laws such as associative law, commutative law, distributive law etc. This powerful algebraic structure with its finiteness makes itself a concise and beautiful subject of study in the mathematical world. With its effective applications in a lot of theories and practice, finite fields becomes gradually an important, well-recognized, algebraic tool in mathematics. Consequently, finite fields attracts the strong interest of mathematicians.The formation and development of finite fields theory has a long history. On the basis of abundant material, the present paper tries to summarize and analyze the history of idea of finite fields in a systematic way. Centering on the development of finite fields, this paper illuminates its relationship to abstract fields and expounds the ideological connections among the mathematicians who had contributed to the theory; at the same time, this paper presents its historical development from the professional viewpoint of the structure of finite fields; finally, this paper introduces some applications of finite fields theory. Due to the limited space and the author's field of vision, the present paper can not cover every aspect of finite fields theory, and can only emphasizes particularly on its application to combinatorics.In conclusion, this paper makes an attempt to expound the process of evolution of idea of finite fields through which we have a comprehensive and clearer knowledge of finite fields theory and its relation to abstract fields. Further more, we can get a general command of the development of algebra.