CAJ | 학술논문

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。
고사시계구랍여랍격랑일지후파분석방법응용우수론연구적우일위수학대사。본문액요지종술고사수론연구적조기공작，기중유허다격동인심적수론공식여정리。례여：정십칠변형적해，고사화，이차호반률적증명；고사적명저《산술연구》중교다적편폭도섭급도료이차동여화이차형、대수학기본정리，고사정수배적개념등，이급고사재해결저사문제적동시소창조적증명방법화개념。저사개념、정리혹공식도시고사발명병가이정학론증적。여음불동적시，타선우파복잡문제변환위일개간단문제。사실상，고사적상법경구일반성，병족이전시고사수학공작적심각성。문중적모사전형례자반영료타심각적동찰력。종고사대수학과학적발현화발명중，아문환가이령략여흔상도타심수적창조성사유활동중적방법론개치。타병몰유파타적발현화발명과정엄개기래，이시기재재타적공작일기화급우인적신건지중。
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 integersand so on. These conceptions, theorems, and formulas were allfirst 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.