Quadratic Irrationals: An Introduction to Classical Number Theory (Chapman & Hall/CRC Pure and Applied Mathematics)

**Quadratic Irrationals: An Introduction to Classical Number Theory** gives a unified treatment of the classical theory of quadratic irrationals. Presenting the material in a modern and elementary algebraic setting, the author focuses on equivalence, continued fractions, quadratic characters, quadratic orders, binary quadratic forms, and class groups. The book highlights the connection between Gauss’s theory of binary forms and the arithmetic of quadratic orders. It collects essential results of the theory that have previously been difficult to access and scattered in the literature, including binary quadratic Diophantine equations and explicit continued fractions, biquadratic class group characters, the divisibility of class numbers by 16, F. Mertens’ proof of Gauss’s duplication theorem, and a theory of binary quadratic forms that departs from the restriction to fundamental discriminants. The book also proves Dirichlet’s theorem on primes in arithmetic progressions, covers Dirichlet’s class number formula, and shows that every primitive binary quadratic form represents infinitely many primes. The necessary fundamentals on algebra and elementary number theory are given in an appendix. Research on number theory has produced a wealth of interesting and beautiful results yet topics are strewn throughout the literature, the notation is far from being standardized, and a unifying approach to the different aspects is lacking. Covering both classical and recent results, this book unifies the theory of continued fractions, quadratic orders, binary quadratic forms, and class groups based on the concept of a quadratic irrational. This Work Focuses On The Number Theory Of Quadratic Irrationalities In Various Forms, Including Continued Fractions, Orders In Quadratic Number Fields, And Binary Quadratic Forms. It Presents Classical Results Obtained By The Famous Number Theorists Gauss, Legendre, Lagrange, And Dirichlet. Collecting Information Previously Scattered In The Literature, The Book Covers The Classical Theory Of Continued Fractions, Quadratic Orders, Binary Quadratic Forms, And Class Groups Based On The Concept Of A Quadratic Irrational-- Chapter 1 Quadratic Irrationals 1 -- 1.1 Quadratic Irrational̀s, Quadratic Number Fields And Discriminants 1 -- 1.2 The Modular Group 7 -- 1.3 Reduced Quadratic Irrationals 16 -- 1.4 Two Short Tables Of Class Numbers 21 -- Chapter 2 Continued Fractions 25 -- 2.1 General Theory Of Continued Fractions 25 -- 2.2 Continued Fractions Of Quadratic Irrationals I: General Theory 38 -- 2.3 Continued Fractions Of Quadratic Irrationals Ii: Special Types 50 -- Chapter 3 Quadratic Residues And Gauss Sums 63 -- 3.1 Elementary Theory Of Power Residues 63 -- 3.2 Gauss And Jacobi Sums 67 -- 3.3 The Quadratic Reciprocity Law 72 -- 3.4 Sums Of Two Squares 79 -- 3.5 Kronecker And Quadratic Symbols 82 -- Chapter 4 L-series And Dirichlet's Prime Number Theorem 99 -- 4.1 Preliminaries And Some Elementary Cases 99 -- 4.2 Multiplicative Functions 102 -- 4.3 Dirichlet L-functions And Proof Of Dirichlet's Theorem 106 -- 4.4 Summation Of L-series 112 -- Chapter 5 Quadratic Orders 115 -- 5.1 Lattices And Orders In Quadratic Number Fields 115 -- 5.2 Units In Quadratic Orders 121 -- 5.3 Lattices And (invertible) Fractional Ideals In Quadratic Orders 129 -- 5.4 Structure Of Ideals In Quadratic Orders 132 -- 5.5 Class Groups And Class Semigroups 140 -- 5.6 Ambiguous Ideals And Ideal Classes 147 -- 5.7 An Application: Some Binary Diophantine Equations 160 -- 5.8 Prime Ideals And Multiplicative Ideal Theory 174 -- 5.9 Class Groups Of Quadratic Orders 179 -- Chapter 6 Binary Quadratic Forms 191 -- 6.1 Elementary Definitions And Equivalence Relations 191 -- 6.2 Representation Of Integers 199 -- 6.3 Reduction 210 -- 6.4 Composition 213 -- 6.5 Theory Of Genera 221 -- 6.6 Ternary Quadratic Forms 240 -- 6.7 Sums Of Squares 248 -- Chapter 7 Cubic And Biquadratic Residues 257 -- 7.1 The Cubic Jacobi Symbol 257 -- 7.2 The Cubic Reciprocity Law 263 -- 7.3 The Biquadratic Jacobi Symbol 271 -- 7.4 The Biquadratic Reciprocity Law 279 -- 7.5 Rational Biquadratic Reciprocity Laws 289 -- 7.6 A Biquadratic Class Group Character And Applications 306 -- Chapter 8 Class Groups 321 -- 8.1 The Analytic Class Number Formula 322 -- 8.2 L-functions Of Quadratic Orders 332 -- 8.3 Ambiguous Classes And Classes Of Order Divisibility By 4 340 -- 8.4 Discriminants With Cyclic 2-class Group: Divisibility By 8 And 16 345. Franz Halter-koch, University Of Graz, Austria. Includes Bibliographical References (p. 407-410) And Index. Quadratic Irrationals Quadratic irrationals, quadratic number fields and discriminants The modular group Reduced quadratic irrationals Two short tables of class numbers Continued Fractions General theory of continued fractionsContinued fractions of quadratic irrationals I: General theory Continued fractions of quadratic irrationals II: Special types Quadratic Residues and Gauss Sums Elementary theory of power residues Gauss and Jacobi sums The quadratic reciprocity law Sums of two squares Kronecker and quadratic symbols L-Series and Dirichlet's Prime Number Theorem Preliminaries and some elementary cases Multiplicative functions Dirichlet L-functions and proof of Dirichlet's theorem Summation of L-series Quadratic Orders Lattices and orders in quadratic number fields Units in quadratic orders Lattices and (invertible) fractional ideals in quadratic orders Structure of ideals in quadratic orders Class groups and class semigroups Ambiguous ideals and ideal classes An application: Some binary Diophantine equations Prime ideals and multiplicative ideal theory Class groups of quadratic ordersBinary Quadratic Forms Elementary definitions and equivalence relationsRepresentation of integers ReductionComposition Theory of genera Ternary quadratic forms Sums of squaresCubic and Biquadratic Residues The cubic Jacobi symbol The cubic reciprocity law The biquadratic Jacobi symbol The biquadratic reciprocity law Rational biquadratic reciprocity laws A biquadratic class group character and applicationsClass Groups The analytic class number formula L-functions of quadratic orders Ambiguous classes and classes of order divisibility by 4 Discriminants with cyclic 2-class group: Divisibility by 8 and 16 Appendix A: Review of Elementary Algebra and Number Theory Appendix B: Some Results from Analysis Bibliography List of Symbols Subject Index