Quadratics

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Overview

The first thing you will find out about this book is that it is fun to read. It is meant for the browser, as well as for the student and for the specialist wanting to know about the area. The footnotes give an historical background to the text, in addition to providing deeper applications of the concept that is being cited. This allows the browser to look more deeply into the history or to pursue a given sideline. Those who are only marginally interested in the area will be able to read the text, pick up information easily, and be entertained at the same time by the historical and philosophical digressions. It is rich in structure and motivation in its concentration upon quadratic orders.

This is not a book that is primarily about tables, although there are 80 pages of appendices that contain extensive tabular material (class numbers of real and complex quadratic fields up to 104; class group structures; fundamental units of real quadratic fields; and more!). This book is primarily a reference book and graduate student text with more than 200 exercises and a great deal of hints!

The motivation for the text is best given by a quote from the Preface of Quadratics: "There can be no stronger motivation in mathematical inquiry than the search for truth and beauty. It is this author's long-standing conviction that number theory has the best of both of these worlds. In particular, algebraic and computational number theory have reached a stage where the current state of affairs richly deserves a proper elucidation. It is this author's goal to attempt to shine the best possible light on the subject."

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Editorial Reviews

Booknews
Presents detailed information on the infrastructure of quadratic order and the continued fraction approach, for graduate students learning algebraic and computational number theory, and for advanced undergraduates with a first course in algebra and number theory. Chapters on prime-producing polynomials; class numbers; and algorithms contain numbered examples, theorems, remarks, lemmas, and definitions, as well as numerous exercises. Appendices offer reference tables, and overviews of classical theory and analytic number theory. Also of interest to research mathematicians and computer scientists. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

Table of Contents

List of Symbols
Preface
Introduction
Background from Algebraic Number Theory
Quadratic Fields: Integers and Units
The Arithmetic of Ideals in Quadratic Fields
The Class Group and Class Number
Reduced Ideals
Quadratic Orders
Powerful Numbers: An Application of Real Quadratics
Continued Fractions Applied to Quadratic Fields
Continued Fractions and Real Quadratics: The Infrastructure
The Continued Fraction Analogue for Complex Quadratics
Diophantine Equations and Class Numbers
Class Numbers and Complex Quadratics
Real Quadratics and Diophantine Equations
Reduced Ideals and Diophantine Equations
Class Numbers and Real Quadratics
Halfway to a Solution
Prime-Producing Polynomials
Complex Prime-Producers
Real Prime-Producers
Density of Primes
Class Numbers: Criteria and Bounds
Factoring Rabinowitsch
Class Number One Criteria
Class Number Bounds via the Divisor Function
The GRH: Relevance of the Riemann Hypothesis
Ambiguous Ideals
Ambiguous Cycles in Real Orders: The Palindromic Index
Exponent Two
Influence of the Infrastructure
Quadratic Residue Covers
Consecutive Powers
Algorithms
Computation of the Class Number of a Real Quadratic Field
Cryptology
Implications of Computational Mathematics for the Philosophy of Mathematics
Appendix A: Tables
Table A1: This is a list of all positive fundamental radicands with class number hΔ = 1 and period length l , of the simple continued fraction expansion of the principal class, less then 24. Table A8 is known to be unconditionally complete whereas Table A1 is complete with one GRH-ruled out exception, as are Tables A2-A4, A6-A7 and A9
Table A2: This is a subset of Table A1 with D - 1 (mod 8)
Table A3: hΔ = 2 for fundamental radicands D > 0 with l < 24
Table A4: This is a list of all fundamental radicands of ERD-type with class groups of exponent 2, broken down into three parts depending on congruence modulo 4 of the radicand
Table A5: This three-part table is an illustration of a computer run done for the proof of Theorem 6.2.2
Table A6: This is a list of all fundamental radicands D > 0 of ERD-type having no split primes less than the Minkowski bound
Table A7: This is a complete list of all fundamental radicands D > 0 with nΔ ≠ 0 (see Exercise 3.2.11) and associated regulators, such that the class number is 1
Table A8: This is a list of all fundamental discriminants D - 1 (mod 8) of ERD-type with class number less than 24, and is known to be unconditionally complete
Table A9: This table lists all fundamental discriminants of ERD-type with class number 2
Appendix B: Fundamental Units of Real Quadratic Fields
This list is broken up into three parts according to congruence modulo 4 of fundamental radicands less than 2 . 103
Appendix C: Class Numbers of Real Quadratic Fields
This table is presented in matrix form with each entry describing a specified class number together with the norm of the fundamental unit with radicands less than 104
Appendix D: Class Numbers of Complex Quadratic Fields (and their class group structure)
This is a table of fundamental radicands D

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