Quadratics: 1st Edition (Hardback) book cover

Quadratics

1st Edition

By Richard A. Mollin

CRC Press

420 pages

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Hardback: 9780849339837
pub: 1995-10-24
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pub: 2018-04-27
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Description

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."

Reviews

"A great deal to be welcomed in this book…It collects much information in one of the major areas of modern computational number theory…Numerous exercises and problems make it a foundation for various postgraduate courses."

- Roger Cook, London Mathematical Society

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 h? = 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: h? = 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 n? ? 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

About the Series

Discrete Mathematics and Its Applications

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Subject Categories

BISAC Subject Codes/Headings:
COM053000
COMPUTERS / Security / General
MAT022000
MATHEMATICS / Number Theory
MAT036000
MATHEMATICS / Combinatorics