A Historical Survey On The Congruent Theory Of Gauss's Disquisitiones Arithmeticae

Gauss's Disquisitiones Arithmeticae is a classic work in the history of number theory. It marked the beginning of modern number theory. The congruent theory is one of the core contents of number theory, containing a great deal of special thought, concept and method. The thoughts of the congruent theory are not studied systematically in Chinese and western works until now. Some correlative contexts often appear in the comprehensive books only, most of which are brief introductions. It is difficult to understand its essence for the reader. Owing to the importance of Disquisitiones Arithmeticae in histoiy of number theory and the core position of the congruent theory in elementary number theory, this paper is to focus on the following two topics: the origin and development of Fermat's little theorem and the quadratic reciprocity law which was hailed as golden by Gauss.Main work follows:(1) In the first part of this paper, a historical development of the number theory before Gauss is reviewed.Based on the systematic analysis of Gauss's work in science and mathematics, inquiry into the mathematical background that Disquisitiones Arithmeticae appeals and Gauss's congruent theory;(2) The development process of Fermat's little theorem and its important function in the compositeness test is elaborated through original literature.we think that the first three section of Disquisitiones Arithmeticae is a summary and development for ancestors' work about Fermat's little theorem,show that Fermat's little theorem played an important role in the elementary number theory;(3) With the two main sources of the quadratic reciprocity law, investigating Fermat,Euler,Lagrange,Legendre, until the related work of Gauss,the way to realize the law's huge push to the development of algebraic number theory in 19 centuries.By reviewing carefully original literature,it is pointed out that pursing a more general reciprocity law maby the most motive of the development of algebraic number theory, usual material mainly emphasizes the function of Fermat's last theorem.