Quadratic Residues and Non-Residues: Selected Topics

by Steve Wright

Overview

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.

Product Details

ISBN-13:	9783319459547
Publisher:	Springer International Publishing
Publication date:	12/13/2016
Series:	Lecture Notes in Mathematics , #2171
Edition description:	1st ed. 2016
Pages:	292
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

After earning degrees in mathematics from Western Kentucky University and Indiana University, the author joined the faculty at Oakland University, where he is now Professor of Mathematics in the Department of Mathematics and Statistics. He currently occupies his time studying number theory.

Table of Contents

Chapter 1. Introduction: Solving the General Quadratic Congruence Modulo a Prime.- Chapter 2. Basic Facts.- Chapter 3. Gauss' Theorema Aureum: the Law of Quadratic Reciprocity.- Chapter 4. Four Interesting Applications of Quadratic Reciprocity.- Chapter 5. The Zeta Function of an Algebraic Number Field and Some Applications.- Chapter 6. Elementary Proofs.- Chapter 7. Dirichlet L-functions and the Distribution of Quadratic Residues.- Chapter 8. Dirichlet's Class-Number Formula.- Chapter 9. Quadratic Residues and Non-residues in Arithmetic Progression.- Chapter 10. Are quadratic residues randomly distributed?.- Bibliography.