互反律

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 laws huge push to the development of algebraic number theory in 19 centuries.

本文主要做了以下工作：(1)首先回顾了高斯之前的数论研究状况，在系统分析高斯的科学与数学成就的基础上，探讨了《算术研究》出现的数学背景和高斯的同余理论；(2)通过对原始文献的系统解读，深入分析了费马小定理发现发展的历程以及在素性检验中的重要作用，指出《算术研究》前三节是高斯在总结并发展了前人对该定理研究的基础上形成的，并揭示了费马小定理在初等数论定理证明中的核心地位；(3)以二次互反律的两个主要来源为线索，详细考察了费马，欧拉，拉格朗目，勒让德，直到高斯的相关工作，揭示了该定律对十九世纪数论发展的巨大推动作用。

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 Fermats little theorem and the quadratic reciprocity law which was hailed as golden by Gauss.

鉴于《算术研究》在数论发展史上的重要性以及同余理论在初等数论中的核心地位，本文重点研究费马小定理和被高斯誉为"黄金定律"的二次互反律的起源和发展。

The first computational result on Gauss sums is given by Gauss himself around 1800 and is applied to the famous Gauss quadratic reciprocity law.

高斯和的第一个计算结果是由高斯于1800年给出，用来研究他的著名的二次互反律。

Quadratic reciprocity.

明了数理定理——二次互反律。

This proof is quite diffent from the classic ones and can be used to study the Iwasawa theory of crystalline representations.

这一证明不同于经典的证明精确互反律的方法，并且可以用于研究晶体表示的Iwasawa理论。

And Pharaoh spoke to Joseph, saying, Your father and your brothers have come to you.

47：5 法老对约瑟说，你父亲和你弟兄们到你这里来了。

Additionally, the approximate flattening of surface strip using lines linking midpoints on perpendicular lines between geodesic curves and the unconditional extreme value method are discussed.

提出了用测地线方程、曲面上两点间短程线来计算膜结构曲面测地线的方法，同时，采用测地线间垂线的中点连线和用无约束极值法进行空间条状曲面近似展开的分析。