The first thing readers will notice about this book is that it is fun to read. It is meant for browsers, for students, and for specialists wanting to know more about the subject. The footnotes give an historical background to the text, in addition to providing deeper applications of the concept that is being cited. This allows browsers 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.

Zusammenfassung

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."

Inhalt

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