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This book is an account of the classical theory of quadratic residues and non-residues with the goal of using that theory as a window 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 cover the basic facts and some of the history about quadratic residues and non-residues and discuss in depth various proofs of the Law of Quadratic Reciprosity, with an emphasis on the six proofs that Gauss published. The remaining seven chapters deal withsome 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 will be of interest to graduate and advanced undergraduate students and to mathematicians interested in number theory.
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