Bilinear algebra. An introduction to the algebraic theory of quadratic forms.

*(English)*Zbl 0890.11011
Algebra, Logic and Applications. 7. Langhorne, PA: Gordon and Breach Science Publishers. xii, 486 p. (1997).

This book is addressed to undergraduate or beginning graduate students. It is a fine textbook which covers the elementary parts of bilinear algebra and the theory of quadratic forms. Thus it can also be used as an introductory text to the more advanced books of T. Y. Lam, Milnor-Husemoller and Scharlau.

Throughout the book the presentation is clear and precise. Full proofs are given, and side-results or different versions of a theorem are included whenever this allows a new view or fuller understanding of the subject. Every section contains 10 or more well-chosen exercises at the end.

To give an impression of the contents of the book let me state the headings of the sections. Part I: Bilinear spaces. 1 Introduction, 2 Bilinear spaces, 3 Bases and matrices of bilinear spaces, 4 Isometries of bilinear spaces, 5 Nonsingular bilinear spaces, 6 Diagonalization of bilinear spaces, 7 Witt’s cancellation theorem, 8 Witt’s chain isometry theorem, 9 Symmetric spaces over some fields, 10 Isometry groups.

Part II: Witt rings. 11 Metabolic and hyperbolic spaces, 12 Witt decomposition of symmetric spaces, 13 Witt group, 14 Tensor products, 15 Witt ring, 16 Quadratic forms, 17 Pfister forms, 18 Formally real fields and ordered fields, 19 Prime ideals of the Witt ring, 20 Witt equivalence of fields.

Part III: Invariants. 21 Algebras, 22 Quaternion algebras, 23 Tensor product of algebras, 24 Brauer group, 25 Hasse and Witt invariants.

Appendices. A Symbolic Hasse and Witt invariants, B Selected problems. Bibliography: 90 items.

On the whole this is a very welcome addition to the existing literature, especially for teaching reasons.

Throughout the book the presentation is clear and precise. Full proofs are given, and side-results or different versions of a theorem are included whenever this allows a new view or fuller understanding of the subject. Every section contains 10 or more well-chosen exercises at the end.

To give an impression of the contents of the book let me state the headings of the sections. Part I: Bilinear spaces. 1 Introduction, 2 Bilinear spaces, 3 Bases and matrices of bilinear spaces, 4 Isometries of bilinear spaces, 5 Nonsingular bilinear spaces, 6 Diagonalization of bilinear spaces, 7 Witt’s cancellation theorem, 8 Witt’s chain isometry theorem, 9 Symmetric spaces over some fields, 10 Isometry groups.

Part II: Witt rings. 11 Metabolic and hyperbolic spaces, 12 Witt decomposition of symmetric spaces, 13 Witt group, 14 Tensor products, 15 Witt ring, 16 Quadratic forms, 17 Pfister forms, 18 Formally real fields and ordered fields, 19 Prime ideals of the Witt ring, 20 Witt equivalence of fields.

Part III: Invariants. 21 Algebras, 22 Quaternion algebras, 23 Tensor product of algebras, 24 Brauer group, 25 Hasse and Witt invariants.

Appendices. A Symbolic Hasse and Witt invariants, B Selected problems. Bibliography: 90 items.

On the whole this is a very welcome addition to the existing literature, especially for teaching reasons.

Reviewer: A.Pfister (Mainz)

##### MSC:

11Exx | Forms and linear algebraic groups |

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

11E04 | Quadratic forms over general fields |

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

12D15 | Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) |

11E39 | Bilinear and Hermitian forms |

15A63 | Quadratic and bilinear forms, inner products |