1st Edition

Quadratic Irrationals An Introduction to Classical Number Theory

By Franz Halter-Koch Copyright 2013
    432 Pages
    by Chapman & Hall

    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.

    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
    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


    List of Symbols

    Subject Index


    Franz Halter-Koch retired as a professor of mathematics from the University of Graz in 2004. A member of the Austrian Academy of Science, Dr. Halter-Koch is the author/coauthor of roughly 150 scientific articles, author of Ideal Systems: An Introduction to Multiplicative Ideal Theory, and coauthor of Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory. His research focuses on elementary and algebraic number theory, non-unique factorizations, and abstract multiplicative ideal theory.

    "… [a] successful attempt to present a cohesive treatment of several important topics in classical number theory by first developing a theory of quadratic irrationals and then building on that to show how the other topics reflect different faces of this theory. … works best as a monograph for those who are already familiar with some parts of the material covered here and would like to see other approaches. … it does include a better selection of numerical examples than is found in most books, and it has a number of applications to other areas such as diophantine equations."
    —Allen Stenger, MAA Reviews, January 2014

    "…comprehensive text dealing with the theory of binary quadratic forms. … one of the main interests of this book is that it gathers a lot of important results which are generally strewn throughout the literature and are quite difficult to access. This text is quite clear and some exercises and examples are scattered throughout the book. I therefore think that it will find many interested readers within the international mathematical community."
    — Olivier Bordellès, Mathematical Reviews Clippings, December 2013