高斯数论研究刍议及其生平补遗——纪念高斯逝世160周年  

Gauss number theory studies and his life addendum

作　　者：金亚南[1] 徐沥泉 

机构地区：[1]无锡旅游商贸高等职业技术学校,江苏无锡214045 [2]无锡市教育科学研究院,江苏无锡214001

出　　处：《自然杂志》2015年第5期348-354,共7页Chinese Journal of Nature

摘　　要：高斯是继欧拉与拉格郎日之后把分析方法应用于数论研究的又一位数学大师。本文扼要地综述高斯数论研究的早期工作,其中有许多激动人心的数论公式与定理。例如:正十七边形的解,高斯和,二次互反律的证明;高斯的名著《算术研究》中较多的篇幅都涉及到了二次同余和二次型、代数学基本定理,高斯整数环的概念等,以及高斯在解决这些问题的同时所创造的证明方法和概念。这些概念、定理或公式都是高斯发明并加以精确论证的。与众不同的是,他善于把复杂问题变换为一个简单问题。事实上,高斯的想法更具一般性,并足以展示高斯数学工作的深刻性。文中的某些典型例子反映了他深刻的洞察力。从高斯对数学科学的发现和发明中,我们还可以领略与欣赏到他深邃的创造性思维活动中的方法论价值。他并没有把他的发现和发明过程掩盖起来,而是记载在他的工作日记和给友人的信件之中。As we know, Carl Friedrich Gauss was a mathematician to make use of mathematical analysis to research the number theory after Euler and Lagrange. An introduction of his study is presented systematically here. There are many exciting formulas and theorems such as the constructability of the regular 17-gon, Gaussian sum and the law of quadratic reciprocity. His main number- theoretical work, Disquisitions Arithmeticae, and several smaller number-theoretical papers contain so many deep and technical results that Fundamental Theorem of Algebra and the ring of Guassian integers and so on. These conceptions, theorems, and formulas were all first discovered accurately by Gauss＇ demonstrations. Gauss was extraordinary at converting a complex question into a simple problem. In fact, Gauss＇ ideas have become more generalized. These facts are enough to prove that he had extensive and deep knowledge of his subject. A few instances represent his deep insight. Besides, we can appreciate the basic principle of methodology from Gauss＇ inventions and discoveries. He never takes a process of discovery in a cover-up, and we know this from his diary which informs us about his most important discoveries.

关 键 词：算术研究 正十七边形 二次互反律 高斯和 高斯整数环 

分 类 号：O156[理学—数学]