To the Gauss Reciprocity Law

Abstract

This chapter consists of elementary number theory and deals with the greatest common divisor, the euclidean algorithm, congruences, linear equations, primitive roots, and the quadratic reciprocity law. The material covered here corresponds to the first four chapters of Gauss’s Disquisitiones arithmeticae (1801) and to the whole volume (70 pages) of Weil’s Number Theory for Beginners (1985). Equally small (95 pages) Bakers A Concise Introduction to the Theory of Numbers (1984) contains, in addition to those standards, quadratic forms, diophantine approximation, Fermat primes, Mersenne primes, Goldbach’s conjecture, twin primes, perfect numbers, the Riemann hypothesis, Euler’s constant, ζ(2n + 1), Fermat’s conjecture, Catalan’s conjecture, and so on. Following Weil, we regard this part of number theory as a rich source of structures of algebra such as groups, rings, and fields. Like computers in these days, the algebraic language is useful and economizes our thoughts. For example, group theory teaches us that the famous Fermat-Euler theorem on congruences is a special case of a simple theorem on finite groups and that the important existence theorem of primitive roots due to Gauss follows from the theorem stating that a finite subgroup of the multiplicative group of any field is cyclic. By doing so, number theory (elementary or advanced) is well ventilated. As for the proof of the reciprocity law, we followed the method of Gauss using Gauss sums because it is not only beautiful but also has a crucial influence on the later development of algebraic number theory.