Quadratic Residues and Non-Residues: Selected Topics

Description

This book offers an account of the classical theory of quadratic residues and non-residues with the goal of using that theory as a lens through which to view the development of some of the fundamental methods employed in modern elementary, algebraic, and analytic number theory. The first three chapters present some basic facts and the history of quadratic residues and non-residues and discuss various proofs of the Law of Quadratic Reciprosity in depth, with an emphasis on the six proofs that Gauss published. The remaining seven chapters explore some interesting applications of the Law of Quadratic Reciprocity, prove some results concerning the distribution and arithmetic structure of quadratic residues and non-residues, provide a detailed proof of Dirichlet’s Class-Number Formula, and discuss the question of whether quadratic residues are randomly distributed. The text is a valuable resource for graduate and advanced undergraduate students as well as for mathematicians interested in number theory. Front Matter....Pages i-xiii Introduction: Solving the General Quadratic Congruence Modulo a Prime....Pages 1-8 Basic Facts....Pages 9-19 Gauss’ Theorema Aureum: The Law of Quadratic Reciprocity....Pages 21-77 Four Interesting Applications of Quadratic Reciprocity ....Pages 79-118 The Zeta Function of an Algebraic Number Field and Some Applications....Pages 119-150 Elementary Proofs....Pages 151-160 Dirichlet L-Functions and the Distribution of Quadratic Residues....Pages 161-201 Dirichlet’s Class-Number Formula....Pages 203-226 Quadratic Residues and Non-Residues in Arithmetic Progression....Pages 227-271 Are Quadratic Residues Randomly Distributed?....Pages 273-283 Back Matter....Pages 285-294