Invitation to quadratic forms.
(Invitation aux formes quadratiques.)

*(French)*Zbl 1235.11003
Mathématiques en Devenir 104. Paris: Calvage et Mounet (ISBN 978-2-91-635219-0/pbk). xxiii, 875 p. (2010).

The central topic of the book under review is the theory of quadratic forms on finite-dimensional vector spaces over a field \(K\).

This subject, that is, the study of homogeneous polynomials of degree two, has a long history in both geometry and algebra, with many fascinating aspects and fundamental applications. In this context, the present book provides a very profound, versatile and thorough introduction to the algebraic and geometric theory of quadratic forms over a field, which appears to be perfectly suited as a course book for seasoned students as well as a textbook for comprehensive self-study.

As for the precise contents of the present voluminous text, the material is organized in five major parts titled as follows:

Part 1. Elementary theory of quadratic forms;

Part 2. Geometry and quadratic forms;

Part 3. Introduction to algebraic theory of quadratic forms;

Part 4. Orthogonal groups, Clifford algebras and spinor groups; Part 5. Quadratic forms in characteristic 2.

Each part comprises several chapters, each of which is subdivided into a number of sections. Altogether, there are thirty-five chapters and six appendices forming the current textbook.

After a very instructive preface, in which the author very thoroughly explains both the concern and the structure of the entire book, together with a few historical remarks sketching the development of the theory of quadratic forms in the last two hundred years, Part 1 lays the foundation for the rest of the book. More precisely, the nine chapters constituting this part provide the general basic material on quadratic forms, including (1) the relevant matrix calculus, (2) bilinear forms and their associated quadratic forms, (3) invariants associated to quadratic forms, (4) the orthogonal decomposition of a bilinear form, (5) the diagonalization of a quadratic form, (6) complex and real quadratic forms, (7) isotropy and hyperbolic spaces, (8) the theorems of Witt, and (9) Witt equivalence and the classification of quadratic forms.

Part 2 contains the following five chapters and is devoted to the geometric aspects of quadratic forms, that is, to the study of affine and projective quadrics. In the course of this part, the reader gets acquainted with (10) Euclidean spaces and their endomorphisms, (11) the structure of the orthogonal group \(O_n(\mathbb R)\), (12) projective and affine quadrics, (13) proper quadrics, and (14) projective conics.

Part 3 turns to the more topical theme of the algebraic theory of quadratic forms, with Witt groups and Witt rings being the central objects of discussion. This part incorporates six further chapters treating the following topics respectively: (15) tensor products of quadratic forms and base change, (16) the Witt group and the Witt-Grothendieck group, (17) the Witt group \(W(\mathbb Q)\), (18) \(p\)-adic quadratic forms and the Hasse principle, (19) the Witt ring and the Witt invariant, and (20) multiplicative forms and Pfister forms.

Containing the subsequent nine chapters, Part 4 offers an extended study of the orthogonal group of a non-degenerate quadratic form, thereby illuminating the fascinating interplay between Witt theory, Clifford algebras, spinor geometry, the Brauer group, and other constructions. The topics discussed in this part are: (21) the structure of the orthogonal group, (22) the spinor norm, (23) \(\mathbb Z/2\)-graded algebras, (24) Clifford algebras, (25) calculus of Clifford algebras, (26) the theory of spinor groups, (27) spinor groups in dimensions 2, 3 and 4, (28) spinor groups in dimension 5 and 6, (29) the Brauer group and the Clifford invariant.

Part 5 of the book is devoted to the special case of quadratic forms over a field of characteristic 2, which had to be excluded in the previous chapters. The peculiarities of this highly exceptional case are thoroughly explained in the remaining six chapters of the book, in which the following themes are touched upon: (30) introduction to quadratic forms in characteristic 2, (31) alternating forms and the symplectic group, (32) regular quadratic forms in characteristic 2 and the Arf invariant, (33) \(\text{Sp}_5(\mathbb F_2)\) and quadratic forms of dimension 4 over \(\mathbb F_2\), (34) Clifford algebras in characteristic 2 and the Dickson invariant, (35) symmetric bilinear forms in characteristic 2 and their classification.

These thirty-five chapters are accompanied by six appendices, in which some basic background material is compiled for the convenience of the reader. This includes: (A) complements of linear algebra, (B) complements of group theory, (C) the Legendre symbol, (D) elements of projective geometry, (E) structure of quaternions, and (F) tensor products of vector spaces and algebras.

Also, each of the thirty-five chapters comes with a large number of exercises and working problems, which are arranged according to the respective material covered in the single sections of a chapter. In fact, the exercises and working problems are extremely numerous, versatile and thoughtfully selected, which certainly must be seen as one of the outstanding features of the present textbook. It is fair to say that this wealth of exercises provides a corresponding wealth of additional material for the self-reliant reader, perhaps even to the extent of an accompanying, further-leading textbook on the subject and its allied theories.

The presentation of the core material captivates by its lucidity, profundity, panoramic width, and expository skill likewise.

No doubt, the author has provided an excellent introduction to various aspects of the theory of quadratic forms over a field, thereby exhibiting many of its beautiful facets in a truly inviting manner. Moreover, tailor-made for the needs of students, the book under review should be seen as a particularly useful introduction to the standard texts in the field, notably to the venerable classics by T. Y. Lam [The algebraic theory of quadratic forms. Mathematics Lecture Note Series. Reading, Mass.: W. A. Benjamin, Inc. Advanced Book Program (1973; Zbl 0259.10019)], W. Scharlau [Quadratic and Hermitian forms. Grundlehren der Mathematischen Wissenschaften, 270. Berlin etc.: Springer-Verlag (1985; Zbl 0584.10010)], A. Pfister [Quadratic forms with applications to algebraic geometry and topology. London Mathematical Society Lecture Note Series. 217. Cambridge: Cambridge Univ. Press (1995; Zbl 0847.11014)], or to the more recent, highly advanced treatise “The algebraic and geometric theory of quadratic forms” by R. Elman, N. Karpenko and A. Merkurjev [The algebraic and geometric theory of quadratic forms. Colloquium Publications. American Mathematical Society 56. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1165.11042)].

This subject, that is, the study of homogeneous polynomials of degree two, has a long history in both geometry and algebra, with many fascinating aspects and fundamental applications. In this context, the present book provides a very profound, versatile and thorough introduction to the algebraic and geometric theory of quadratic forms over a field, which appears to be perfectly suited as a course book for seasoned students as well as a textbook for comprehensive self-study.

As for the precise contents of the present voluminous text, the material is organized in five major parts titled as follows:

Part 1. Elementary theory of quadratic forms;

Part 2. Geometry and quadratic forms;

Part 3. Introduction to algebraic theory of quadratic forms;

Part 4. Orthogonal groups, Clifford algebras and spinor groups; Part 5. Quadratic forms in characteristic 2.

Each part comprises several chapters, each of which is subdivided into a number of sections. Altogether, there are thirty-five chapters and six appendices forming the current textbook.

After a very instructive preface, in which the author very thoroughly explains both the concern and the structure of the entire book, together with a few historical remarks sketching the development of the theory of quadratic forms in the last two hundred years, Part 1 lays the foundation for the rest of the book. More precisely, the nine chapters constituting this part provide the general basic material on quadratic forms, including (1) the relevant matrix calculus, (2) bilinear forms and their associated quadratic forms, (3) invariants associated to quadratic forms, (4) the orthogonal decomposition of a bilinear form, (5) the diagonalization of a quadratic form, (6) complex and real quadratic forms, (7) isotropy and hyperbolic spaces, (8) the theorems of Witt, and (9) Witt equivalence and the classification of quadratic forms.

Part 2 contains the following five chapters and is devoted to the geometric aspects of quadratic forms, that is, to the study of affine and projective quadrics. In the course of this part, the reader gets acquainted with (10) Euclidean spaces and their endomorphisms, (11) the structure of the orthogonal group \(O_n(\mathbb R)\), (12) projective and affine quadrics, (13) proper quadrics, and (14) projective conics.

Part 3 turns to the more topical theme of the algebraic theory of quadratic forms, with Witt groups and Witt rings being the central objects of discussion. This part incorporates six further chapters treating the following topics respectively: (15) tensor products of quadratic forms and base change, (16) the Witt group and the Witt-Grothendieck group, (17) the Witt group \(W(\mathbb Q)\), (18) \(p\)-adic quadratic forms and the Hasse principle, (19) the Witt ring and the Witt invariant, and (20) multiplicative forms and Pfister forms.

Containing the subsequent nine chapters, Part 4 offers an extended study of the orthogonal group of a non-degenerate quadratic form, thereby illuminating the fascinating interplay between Witt theory, Clifford algebras, spinor geometry, the Brauer group, and other constructions. The topics discussed in this part are: (21) the structure of the orthogonal group, (22) the spinor norm, (23) \(\mathbb Z/2\)-graded algebras, (24) Clifford algebras, (25) calculus of Clifford algebras, (26) the theory of spinor groups, (27) spinor groups in dimensions 2, 3 and 4, (28) spinor groups in dimension 5 and 6, (29) the Brauer group and the Clifford invariant.

Part 5 of the book is devoted to the special case of quadratic forms over a field of characteristic 2, which had to be excluded in the previous chapters. The peculiarities of this highly exceptional case are thoroughly explained in the remaining six chapters of the book, in which the following themes are touched upon: (30) introduction to quadratic forms in characteristic 2, (31) alternating forms and the symplectic group, (32) regular quadratic forms in characteristic 2 and the Arf invariant, (33) \(\text{Sp}_5(\mathbb F_2)\) and quadratic forms of dimension 4 over \(\mathbb F_2\), (34) Clifford algebras in characteristic 2 and the Dickson invariant, (35) symmetric bilinear forms in characteristic 2 and their classification.

These thirty-five chapters are accompanied by six appendices, in which some basic background material is compiled for the convenience of the reader. This includes: (A) complements of linear algebra, (B) complements of group theory, (C) the Legendre symbol, (D) elements of projective geometry, (E) structure of quaternions, and (F) tensor products of vector spaces and algebras.

Also, each of the thirty-five chapters comes with a large number of exercises and working problems, which are arranged according to the respective material covered in the single sections of a chapter. In fact, the exercises and working problems are extremely numerous, versatile and thoughtfully selected, which certainly must be seen as one of the outstanding features of the present textbook. It is fair to say that this wealth of exercises provides a corresponding wealth of additional material for the self-reliant reader, perhaps even to the extent of an accompanying, further-leading textbook on the subject and its allied theories.

The presentation of the core material captivates by its lucidity, profundity, panoramic width, and expository skill likewise.

No doubt, the author has provided an excellent introduction to various aspects of the theory of quadratic forms over a field, thereby exhibiting many of its beautiful facets in a truly inviting manner. Moreover, tailor-made for the needs of students, the book under review should be seen as a particularly useful introduction to the standard texts in the field, notably to the venerable classics by T. Y. Lam [The algebraic theory of quadratic forms. Mathematics Lecture Note Series. Reading, Mass.: W. A. Benjamin, Inc. Advanced Book Program (1973; Zbl 0259.10019)], W. Scharlau [Quadratic and Hermitian forms. Grundlehren der Mathematischen Wissenschaften, 270. Berlin etc.: Springer-Verlag (1985; Zbl 0584.10010)], A. Pfister [Quadratic forms with applications to algebraic geometry and topology. London Mathematical Society Lecture Note Series. 217. Cambridge: Cambridge Univ. Press (1995; Zbl 0847.11014)], or to the more recent, highly advanced treatise “The algebraic and geometric theory of quadratic forms” by R. Elman, N. Karpenko and A. Merkurjev [The algebraic and geometric theory of quadratic forms. Colloquium Publications. American Mathematical Society 56. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1165.11042)].

Reviewer: Werner Kleinert (Berlin)

##### MSC:

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11D09 | Quadratic and bilinear Diophantine equations |

11E39 | Bilinear and Hermitian forms |

11E57 | Classical groups |

11E81 | Algebraic theory of quadratic forms; Witt groups and rings |

11E88 | Quadratic spaces; Clifford algebras |

15A63 | Quadratic and bilinear forms, inner products |

15A66 | Clifford algebras, spinors |