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.
Illustrates how the study of quadratic residues led directly to the development of fundamental methods in elementary, algebraic, and analytic number theoryPresents in detail seven proofs of the Law of Quadratic Reciprocity, with an emphasis on the six proofs which Gauss publishedDiscusses in some depth the historical development of the study of quadratic residues and non-residues