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Quadratic Residues and Non-Residues: Selected Topics 1st ed. 2016 [Minkštas viršelis]

  • Formatas: Paperback / softback, 292 pages, aukštis x plotis: 235x155 mm, weight: 4686 g, 20 Illustrations, black and white; XIII, 292 p. 20 illus., 1 Paperback / softback
  • Serija: Lecture Notes in Mathematics 2171
  • Išleidimo metai: 15-Nov-2016
  • Leidėjas: Springer International Publishing AG
  • ISBN-10: 3319459546
  • ISBN-13: 9783319459547
Kitos knygos pagal šią temą:
Quadratic Residues and Non-Residues: Selected Topics 1st ed. 2016Didesnis paveikslėlis
  • Formatas: Paperback / softback, 292 pages, aukštis x plotis: 235x155 mm, weight: 4686 g, 20 Illustrations, black and white; XIII, 292 p. 20 illus., 1 Paperback / softback
  • Serija: Lecture Notes in Mathematics 2171
  • Išleidimo metai: 15-Nov-2016
  • Leidėjas: Springer International Publishing AG
  • ISBN-10: 3319459546
  • ISBN-13: 9783319459547
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783319459547.html
Raktažodžiai:
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 Dirichlets 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.
1 Introduction: Solving the General Quadratic Congruence Modulo a Prime
1(8)
1.1 Linear and Quadratic Congruences
1(4)
1.2 The Disquisitiones Arithmeticae
5(1)
1.3 Notation, Terminology, and Some Useful Elementary Number Theory
6(3)
2 Basic Facts
9(12)
2.1 The Legendre Symbol, Euler's Criterion, and Other Important Things
9(4)
2.2 The Basic Problem and the Fundamental Problem for a Prime
13(3)
2.3 Gauss' Lemma and the Fundamental Problem for the Prime 2
16(5)
3 Gauss' Theorema Aureum: The Law of Quadratic Reciprocity
21(58)
3.1 What is a Reciprocity Law?
22(3)
3.2 The Law of Quadratic Reciprocity
25(3)
3.3 Some History
28(5)
3.4 Proofs of the Law of Quadratic Reciprocity
33(1)
3.5 A Proof of Quadratic Reciprocity via Gauss' Lemma
34(4)
3.6 Another Proof of Quadratic Reciprocity via Gauss' Lemma
38(2)
3.7 A Proof of Quadratic Reciprocity via Gauss Sums: Introduction
40(1)
3.8 Algebraic Number Theory
41(8)
3.9 Proof of Quadratic Reciprocity via Gauss Sums: Conclusion
49(6)
3.10 A Proof of Quadratic Reciprocity via Ideal Theory: Introduction
55(1)
3.11 The Structure of Ideals in a Quadratic Number Field
56(8)
3.12 Proof of Quadratic Reciprocity via Ideal Theory: Conclusion
64(8)
3.13 A Proof of Quadratic Reciprocity via Galois Theory
72(7)
4 Four Interesting Applications of Quadratic Reciprocity
79(40)
4.1 Solution of the Fundamental Problem for Odd Primes
80(3)
4.2 Solution of the Basic Problem
83(5)
4.3 Sets of Integers Which Are Quadratic Residues of Infinitely Many Primes
88(3)
4.4 Intermezzo: Dirichlet's Theorem on Primes in Arithmetic Progression
91(6)
4.5 The Asymptotic Density of Primes
97(1)
4.6 The Density of Primes Which Have a Given Finite Set of Quadratic Residues
98(9)
4.7 Zero-Knowledge Proofs and Quadratic Residues
107(3)
4.8 Jacobi Symbols
110(3)
4.9 An Algorithm for Fast Computation of Legendre Symbols
113(6)
5 The Zeta Function of an Algebraic Number Field and Some Applications
119(32)
5.1 Dedekind's Ideal Distribution Theorem
120(9)
5.2 The Zeta Function of an Algebraic Number Field
129(8)
5.3 The Zeta Function of a Quadratic Number Field
137(2)
5.4 Proof of Theorem 4.12 and Related Results
139(7)
5.5 Proof of the Fundamental Theorem of Ideal Theory
146(5)
6 Elementary Proofs
151(10)
6.1 Whither Elementary Proofs in Number Theory?
151(1)
6.2 An Elementary Proof of Theorem 5.13
152(6)
6.3 An Elementary Proof of Theorem 4.12
158(3)
7 Dirichlet L-Functions and the Distribution of Quadratic Residues
161(42)
7.1 Positivity of Sums of Values of a Legendre Symbol
162(3)
7.2 Proof of Theorem 7.1: Outline of the Argument
165(1)
7.3 Some Useful Facts About Dirichlet L-Functions
166(3)
7.4 Calculation of a Gauss Sum
169(4)
7.5 Some Useful Facts About Analytic Functions of a Complex Variable
173(3)
7.6 The Convergence of Fourier Series
176(6)
7.7 Proof of Theorems 7.2, 7.3, and 7.4
182(10)
7.8 An Elegant Proof of Lemma 4.8 for Real Dirichlet Characters
192(4)
7.9 A Proof of Quadratic Reciprocity via Finite Fourier Series
196(7)
8 Dirichlet's Class-Number Formula
203(24)
8.1 Some Structure Theory for Dirichlet Characters
204(1)
8.2 The Structure of Real Primitive Dirichlet Characters
205(3)
8.3 Elements of the Theory of Quadratic Forms
208(1)
8.4 Representation of Integers by Quadratic Forms and the Class Number
209(3)
8.5 The Class-Number Formula
212(8)
8.6 The Class-Number Formula and the Class Number of Quadratic Fields
220(3)
8.7 A Character-Theoretic Proof of Quadratic Reciprocity
223(4)
9 Quadratic Residues and Non-Residues in Arithmetic Progression
227(46)
9.1 Long Sets of Consecutive Residues and Non-Residues
228(2)
9.2 Long Sets of Residues and Non-Residues in Arithmetic Progression
230(2)
9.3 Weil Sums and Their Estimation
232(6)
9.4 Solution of Problems 1 and 3
238(4)
9.5 Solution of Problems 2 and 4: Introduction
242(2)
9.6 Preliminary Estimate of qε(p)
244(3)
9.7 Calculation of Σ4(p): Preliminaries
247(2)
9.8 The (B, S)-Signature of a Prime
249(2)
9.9 Calculation of Σ4(p): Conclusion
251(3)
9.10 Solution of Problems 2 and 4: Conclusion
254(4)
9.11 An Interesting Class of Examples
258(11)
9.12 The Asymptotic Density of Π+(a, b)
269(4)
10 Are Quadratic Residues Randomly Distributed?
273(12)
10.1 Irregularity of the Distribution of Quadratic Residues
273(2)
10.2 Detecting Random Behavior Using the Central Limit Theorem
275(2)
10.3 Verifying Random Behavior via a Result of Davenport and Erdos
277(8)
Bibliography 285(4)
Index 289
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.