The first thing which you will find about this book is that it is fun to read. It is meant for the browser (as well as the student wanting to know about this area or the specialist). The footnotes give an historical background to the text as well as giving deeper applications of the concept being cited. Thus, the browser may want to look more deeply into the historical footnotes, or pursue a given sideline for interest sake. Colleagues of mine who are only marginally interested in the area have told me that they find it to be an enjoyable read where they can pick up information easily and be entertained at the same time by the historical, and the philosophical digressions. It is rich in structure and motivation in its concentration (solely) upon quadratic orders.
If you are familiar with CRC publications, then you may think that this is primarily a book of tables. This is NOT the case. Although there are appendices which contain extensive lists (80 pages worth!) of class numbers of real and complex quadratic fields (up to 10^4) and class group structures, as well as fundamental units of real quadratic fields up to 2*10^3, plus much more, this is also a reference text and graduate student text with an extensive list of exercises, coupled with elaborate hints.
If you are one of those people who think that "EVERYTHING IS KNOWN ABOUT QUADRATIC ORDERS", then think again. If you believe that, then you are invited to prove the (more than 40) conjectures which have been left in the book, and develop the theory suggested by the exercises and open questions which have been posed in the text. This is a area wherein many exciting and difficult questions remain unanswered and the structure remains largely untapped. The reader is invited to read and learn about this beautiful area. The infrastructure, which is the coupling of continued fractions and ideal theory has only been developed in the past couple of decades, and has not received the respect which it so richly deserves. This book has been written to remedy this situation, and bring the reader to the frontiers of research on the topic.
The CRC flyer has the following as an excerpt:
This unique text/reference looks at quadratics in various guises such as: quadratic diophantine equations, prime-producing quadratic polynomials, class numbers of quadratic orders, ambiguous ideals in quadratic orders, quadratic residue covers, and algorithms from cryptography based upon ideals in the class group of a complex quadratic field. It overviews the theory of binary quadratic forms, genus theory, and composition as it relates to the ideal theory. The new Palindromic index and Quadratic residue covers are described in detail. Quadratics is filled with exercises and detailed hints, historical footnotes, a list of symbols, and appendices for students, mathematicians and computer scientists.
A quote from the Preface of Quadratics sums up my motivation for writing quadratics, which overrides all others:
"There can be no stronger motivation in mathematical inquiry than the search for truth and beauty. It is this author's longstanding conviction that number theory has the best of both 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 upon the subject."
For those of you who already have a copy of the book:
