Quadratic irrationals : an introduction to classical number theory
- Halter-Koch, Franz, 1944-
- Boca Raton : CRC Press, Taylor & Francis Group 
- Copyright notice
- Physical description
- xvi, 415 pages : illustrations ; 27 cm.
- Monographs and textbooks in pure and applied mathematics ; v. 306.
QA247 .H254 2013
- Unknown QA247 .H254 2013
- Includes bibliographical references (pages 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 fractions Continued 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 orders Binary Quadratic Forms Elementary definitions and equivalence relations Representation of integers Reduction Composition Theory of genera Ternary quadratic forms Sums of squares Cubic 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 applications Class 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.
- (source: Nielsen Book Data)
- Publisher's Summary
- 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.
(source: Nielsen Book Data)
- Publication date
- Copyright date
- Franz Halter-Koch, University of Graz, Austria.
- Pure and applied mathematics : a program of monographs, textbooks, and lecture notes ; v. 306