Quadratic Residues and Non-Residues
Selected Topics

Authors: Steve Wright



Illustrates how the study of quadratic residues led directly to the development of fundamental methods in elementary, algebraic, and analytic number theory

Presents in detail seven proofs of the Law of Quadratic Reciprocity, with an emphasis on the six proofs which Gauss published

Discusses in some depth the historical development of the study of quadratic residues and non-residues

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.

Table of contents

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

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