## Quadratic Residues and Non-Residues: Selected TopicsThis 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. |

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### Contents

1 | |

9 | |

The Law of Quadratic Reciprocity | 20 |

4 Four Interesting Applications of Quadratic Reciprocity | 79 |

5 The Zeta Function of an Algebraic Number Field and Some Applications | 119 |

6 Elementary Proofs | 151 |

7 Dirichlet LFunctions and the Distribution of Quadratic Residues | 161 |

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### Common terms and phrases

algebraic integers algebraic number field algorithm arithmetic progression asymptotic calculate Chap class number class-number formula coefficients complex numbers congruence converges Davenport defined denote the set density determined Dirichlet character discriminant Disquisitiones elements equation equivalence classes equivalence relation Euler fact follows Fourier series fundamental fundamental discriminant Galois Gauss sums Hecke 27 hence ideal class ideal-class group implies infinitely many primes integral basis Jacobi symbols L-functions Law of Quadratic Legendre symbol Lemma minimal polynomial mod q non-principal norm number theory odd prime overlap diagram pairwise disjoint positive integer prime factors prime ideals Problem proof of quadratic proof of Theorem Proposition prove Theorem q mod quadratic forms quadratic number field quadratic reciprocity quadratic residues quotient diagram relatively prime residues and non-residues Sect set of residues solution square square-free Suppose Theorem 3.3 verify zeta function