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**Binary quadratic forms. Classical theory and modern computations.**
*(English)*
Zbl 0698.10013

New York, NY etc.: Springer-Verlag. x, 247 p. DM 74.00 (1989).

Almost two centuries after C. F. Gauss [Disquisitiones Arithmeticae (1801)], a century after G. W. Mathews [Theory of Numbers (Cambridge 1892; JFM 24.0162.01)] and sixty years after L. E. Dickson [Introduction to the Theory of Numbers (Chicago 1929; JFM 55.0092.19, Zbl 0002.01103)] were published, a new book on the theory of binary quadratic forms appears on the market. The difference in the titles between the three great classics of the past and the new book certainly says something about the direction of modern number theory.

Why should a book about binary quadratic forms be published in the late 20th century? The author’s stated goal is: “to provide an update of Mathews’ book with modern notations and ‘correct’ definitions of forms, an update which is a complete discussion of the classical theory of binary quadratic forms and also a survey of modern computations and applications.” The “correct” definition of forms is to allow the middle coefficient to be any integer, instead of requiring it to be even as Gauss and Mathews, but not Dickson, did. This allows a direct connection between the theory of binary quadratic forms and the theory of ideals in quadratic number fields. One reason for the current interest in the subject is that the theory of quadratic forms gives a concrete method for computing the class group structure in quadratic fields.

The first four chapters of the book present the classical theory of binary quadratic forms, including reduction, representation, automorphs and composition. Chapter 5 contains a brief discussion about class number computations, including the problem of determining all discriminants having one class per genus. Chapter 6 establishes the link between binary quadratic forms and quadratic fields. Chapter 7 describes the relationship between forms of fundamental and non-fundamental discriminants. Chapters 8 and 9 are concerned with methods for determining the structure of the class group, as well as related problems, such as Pell’s equation and biquadratic reciprocity laws. The final chapter of the book considers some factoring techniques which are related to binary quadratic forms.

While the book is interesting and generally well written, it contains many typographical errors and a few minor blunders, which tend to mar its beauty. For example, the statements “\(q\equiv 1\) (mod \(q\))” and “\(q\equiv 1\) (mod \(q^ 2)\)” appear on page 150, while the statement \(``2^{(p-1)/4}\equiv 1\) (mod \(p\)) for primes congruent to 1 modulo 16,” appears on page 165. Moreover, the book contains no exercises, making it less attractive for a reading course. However, the author’s obvious enthusiasm for the subject makes the book enjoyable to read. Also, this reviewer shares the author’s view concerning the need for a modern book on the theory of quadratic forms.

Why should a book about binary quadratic forms be published in the late 20th century? The author’s stated goal is: “to provide an update of Mathews’ book with modern notations and ‘correct’ definitions of forms, an update which is a complete discussion of the classical theory of binary quadratic forms and also a survey of modern computations and applications.” The “correct” definition of forms is to allow the middle coefficient to be any integer, instead of requiring it to be even as Gauss and Mathews, but not Dickson, did. This allows a direct connection between the theory of binary quadratic forms and the theory of ideals in quadratic number fields. One reason for the current interest in the subject is that the theory of quadratic forms gives a concrete method for computing the class group structure in quadratic fields.

The first four chapters of the book present the classical theory of binary quadratic forms, including reduction, representation, automorphs and composition. Chapter 5 contains a brief discussion about class number computations, including the problem of determining all discriminants having one class per genus. Chapter 6 establishes the link between binary quadratic forms and quadratic fields. Chapter 7 describes the relationship between forms of fundamental and non-fundamental discriminants. Chapters 8 and 9 are concerned with methods for determining the structure of the class group, as well as related problems, such as Pell’s equation and biquadratic reciprocity laws. The final chapter of the book considers some factoring techniques which are related to binary quadratic forms.

While the book is interesting and generally well written, it contains many typographical errors and a few minor blunders, which tend to mar its beauty. For example, the statements “\(q\equiv 1\) (mod \(q\))” and “\(q\equiv 1\) (mod \(q^ 2)\)” appear on page 150, while the statement \(``2^{(p-1)/4}\equiv 1\) (mod \(p\)) for primes congruent to 1 modulo 16,” appears on page 165. Moreover, the book contains no exercises, making it less attractive for a reading course. However, the author’s obvious enthusiasm for the subject makes the book enjoyable to read. Also, this reviewer shares the author’s view concerning the need for a modern book on the theory of quadratic forms.

Reviewer: C.Parry

### MSC:

11E16 | General binary quadratic forms |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11R23 | Iwasawa theory |

12-02 | Research exposition (monographs, survey articles) pertaining to field theory |

11R11 | Quadratic extensions |

11E12 | Quadratic forms over global rings and fields |