##
**Introduction to quadratic forms over fields.**
*(English)*
Zbl 1068.11023

Graduate Studies in Mathematics 67. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-1095-2/hbk). xxi, 550 p. (2005).

In 1973, T. Y. Lam published [The algebraic theory of quadratic forms (1973; Zbl 0259.10019)] (referred to as ATQF in the sequel, following the author’s own notation in the present book). At the time of publication, ATQF was the first self-contained textbook that focussed exclusively on quadratic forms over arbitrary fields of characteristic \(\neq 2\), a theory which had experienced a surge in activity after Pfister’s groundbreaking work in the 1960s. ATQF brought the reader right up to some of the most recent developments at the time and it established itself as the standard reference on the topic, admired for its style, clarity and substance which eventually won the Leroy P. Steele Prize in Mathematical Exposition in 1982. A revised printing appeared in 1980 (see Zbl 0437.10006).

Since then, various other important books on quadratic forms over fields and rings have been published, most notably W. Scharlau [Quadratic and Hermitian forms, Grundlehren der Mathematischen Wissenschaften 270, Berlin: Springer (1985; Zbl 0584.10010)] which covers much of the same material as ATQF, R. Baeza [Quadratic forms over semilocal rings, Lecture Notes in Mathematics 655, Berlin: Springer (1978; Zbl 0382.10014)], M.-A. Knus [Quadratic and Hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften 294, Berlin: Springer (1991; Zbl 0756.11008)], or A. Pfister’s beautiful little book [Quadratic forms with applications to algebraic geometry and topology, London Mathematical Society Lecture Notes Series 217, Cambridge: Cambridge Univ. Press (1995; Zbl 0847.11014)]. But ATQF has stayed a favorite ever since for everyone who wanted to learn the basics of the theory of quadratic forms over fields or who needed it for reference purposes.

However, in view of the many developments in the theory since the first appearance of ATQF, the author recognized the need for a new version, and the present book is the fruit of his efforts. It is not just an updated and revised version but rather a in some sections completely rewritten and considerably expanded version, albeit very much in the spirit of ATQF. It still focusses exclusively on quadratic forms over fields of characteristic \(\neq 2\) (with the exception on two sections on levels and Pythagoras numbers of rings) and it is more or less completely self-contained. As a consequence, many of the more recent results that are based on deep theories from algebraic geometry, algebraic \(K\)-theory or Galois cohomology (e.g. Voevodsky’s proof of the Milnor conjecture, Merkurjev’s and Izhboldin’s results on the \(u\)-invariant, Vishik’s and Karpenko’s results on function fields and higher Witt indices) are merely stated or hinted at, possibly with a proof of a special case if that could be done within the scope of the book. But the reader interested in these results will have a book at hand which provides the context and builds a solid foundation from which one can embark on a further study of these exciting new developments.

The book consists of thirteen chapters, the first eleven of which correspond to the eleven chapters of ATQF, not only in content but also in title. The style is at times a little more leisurely compared to ATQF without losing any of its clarity. Even in those parts where the present book’s exposition is almost identical to that of ATQF, various additional enlightening explanations or examples are interspersed or proofs are simplified. Each chapter is complemented by a set of exercises, a total of about 280 for the whole book (almost twice as many as in ATQF).

Chapter I introduces the basic notions and results and follows closely ATQF. In Chapter II, the Witt\hbox{(-Grothendieck)} ring is introduced. New is a section on the classification of small Witt rings. Chapter III on quaternion algebras and their norm forms is augmented by a section on linkage of quaternion algebras including Albert’s result on when a biquaternion algebra is a division algebra, and by a section on the characterization of quaternion algebras. Chapter IV introduces central simple (graded) algebras and the Brauer(-Wall) group and is essentially the same as in ATQF. Chapter V on Clifford algebras now includes a section on Steinberg symbols and Milnor’s \(k_2F\), including a proof of the isomorphism of \(k_2F\) and \(I^2F/I^3F\) (originally due to Milnor). Merkurjev’s result that the latter is isomorphic to \(_2\hbox{Br}(F)\) is mentioned but not proved. In Chapter VI on quadratic forms over local and global fields, a subsection on the classification of Witt rings of nonreal fields with four square classes is added. Chapter VII deals with quadratic forms under algebraic extensions. Here, several results have been added on the behaviour of the order of the Witt ring resp. the square class group under quadratic or finite extensions, as well as the Gross-Fischer theorem on obtaining field extensions with prescribed square class group and various results on trace forms. What was previously an appendix to Chapter VII in ATQF has now been considerably extended and become a section on quadratic closures.

Chapter VIII on formally real fields, real-closed fields and Pythagorean fields has undergone substantial changes. Most notably, Euclidean fields are introduced, a proof of the uniqueness of the real closure for an ordered field and a different proof of Pfister’s local-global principle are given as well as Artin-Schreier’s result on subfields of finite index inside an algebraically closed field. Sections on applications to Galois theory, on the Harrison topology, and on preorderings have been added. Chapter IX on quadratic forms under transcendental extensions is more or less unchanged. It contains the results by Cassels-Pfister and the exact sequences concerning the Witt ring of the rational function field in one variable by Milnor and by Scharlau.

In ATQF, function fields of quadratic forms have only shown up very sporadically and implicitly. However, many of the most important recent results in the algebraic theory of quadratic forms concern such function fields. Chapter X, which originally only developed the theory of (strongly) multiplicative and Pfister forms and which, as an application, gave a proof of the Arason-Pfister Hauptsatz, now contains extensive sections on function fields of quadratic forms, addressing among other things the question of isotropy and hyperbolicity of quadratic forms over such function fields, and stating (without proof) some of the recent results by the reviewer, Karpenko, Vishik, Karpenko-Merkurjev and others on questions such as the first higher Witt index, dimensions in \(I^nF\), essential dimension of quadratic forms etc. The chapter also includes a section on the Milnor conjecture.

Chapter XI is concerned with field invariants such as the level, the Pythagoras number and the \(u\)-invariant. This chapter has also undergone substantial changes and now includes a section on the property \(A_n\) and a proof of Leep’s theorem on the possible growth of the \(u\)-invariant under finite extensions.

The last two chapters XII and XIII on special topics in quadratic forms and on invariants are completely new and comprise more than one hundred pages. Without claiming to be a systematic account of specific aspects of the theory, they give however a nice flavor of the various directions in which the theory branches out and of its multifarious facets. Topics include Harrison’s isomorphism criterion for Witt rings, a study of quadratic forms of small dimension, classification of quadratic forms over certain fields, Kaplansky radical, (pre-)Hilbert fields, axiomatic quadratic form theories, the construction of fields with prescribed invariants, the level and Pythagoras number of commutative rings.

At the end of ATQF, Lam listed eight open problems on which much of the research in quadratic forms was focussed at the time. And again, in the present book, Lam closes by posing this time ten open questions that attract current interest or might stimulate further research. These questions pertain mainly to field invariants and their possible values and their behaviour when passing to a purely transcendental extension of a field for which the values are known. One question concerns the isomorphism of function fields of quadratic forms (the “quadratic” Zariski problem). All these questions have deceptively simple formulations, and five of them have already appeared in ATQF (albeit formulated sometimes a little differently due to partial progress toward their solution). It is perhaps an indication of the difficulty of the subject that answers to these questions still elude us more than thirty years after they were asked in ATQF. On the other hand, the many spectacular advances which have taken place over the last thirty years, including solutions to some of the questions asked in ATQF, show that the algebraic theory of quadratic forms over fields has been and continues to be a thriving theory. In fact, the quest for solving some of these questions has necessitated and inspired new developments and the creation of new methods in other areas such as algebraic and arithmetic geometry, Galois cohomology, algebraic \(K\)-theory, central simple algebras, or real algebraic geometry, to name but a few.

The present book is a wonderful achievement. Its genesis is recounted with charm and warmth in the preface. The author’s lucid style and expository skill, his judicious choice of topics and their impeccable layout, not to mention the beautiful typesetting (an obvious improvement compared to the typography of ATQF which was written on an IBM Selectric typewriter) make the book a joy to read or just to browse. Anybody owning a copy of ATQF (most likely worn out and on the verge of disintegrating like the reviewer’s) will certainly want to have a copy of the present book on his or her shelf. But it is sure to attract many new admirers and it will be a must for anybody working in quadratic forms or on topics related to or using quadratic forms, be it for learning the theory of quadratic forms over fields from its foundations, or be it as a reference.

Since then, various other important books on quadratic forms over fields and rings have been published, most notably W. Scharlau [Quadratic and Hermitian forms, Grundlehren der Mathematischen Wissenschaften 270, Berlin: Springer (1985; Zbl 0584.10010)] which covers much of the same material as ATQF, R. Baeza [Quadratic forms over semilocal rings, Lecture Notes in Mathematics 655, Berlin: Springer (1978; Zbl 0382.10014)], M.-A. Knus [Quadratic and Hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften 294, Berlin: Springer (1991; Zbl 0756.11008)], or A. Pfister’s beautiful little book [Quadratic forms with applications to algebraic geometry and topology, London Mathematical Society Lecture Notes Series 217, Cambridge: Cambridge Univ. Press (1995; Zbl 0847.11014)]. But ATQF has stayed a favorite ever since for everyone who wanted to learn the basics of the theory of quadratic forms over fields or who needed it for reference purposes.

However, in view of the many developments in the theory since the first appearance of ATQF, the author recognized the need for a new version, and the present book is the fruit of his efforts. It is not just an updated and revised version but rather a in some sections completely rewritten and considerably expanded version, albeit very much in the spirit of ATQF. It still focusses exclusively on quadratic forms over fields of characteristic \(\neq 2\) (with the exception on two sections on levels and Pythagoras numbers of rings) and it is more or less completely self-contained. As a consequence, many of the more recent results that are based on deep theories from algebraic geometry, algebraic \(K\)-theory or Galois cohomology (e.g. Voevodsky’s proof of the Milnor conjecture, Merkurjev’s and Izhboldin’s results on the \(u\)-invariant, Vishik’s and Karpenko’s results on function fields and higher Witt indices) are merely stated or hinted at, possibly with a proof of a special case if that could be done within the scope of the book. But the reader interested in these results will have a book at hand which provides the context and builds a solid foundation from which one can embark on a further study of these exciting new developments.

The book consists of thirteen chapters, the first eleven of which correspond to the eleven chapters of ATQF, not only in content but also in title. The style is at times a little more leisurely compared to ATQF without losing any of its clarity. Even in those parts where the present book’s exposition is almost identical to that of ATQF, various additional enlightening explanations or examples are interspersed or proofs are simplified. Each chapter is complemented by a set of exercises, a total of about 280 for the whole book (almost twice as many as in ATQF).

Chapter I introduces the basic notions and results and follows closely ATQF. In Chapter II, the Witt\hbox{(-Grothendieck)} ring is introduced. New is a section on the classification of small Witt rings. Chapter III on quaternion algebras and their norm forms is augmented by a section on linkage of quaternion algebras including Albert’s result on when a biquaternion algebra is a division algebra, and by a section on the characterization of quaternion algebras. Chapter IV introduces central simple (graded) algebras and the Brauer(-Wall) group and is essentially the same as in ATQF. Chapter V on Clifford algebras now includes a section on Steinberg symbols and Milnor’s \(k_2F\), including a proof of the isomorphism of \(k_2F\) and \(I^2F/I^3F\) (originally due to Milnor). Merkurjev’s result that the latter is isomorphic to \(_2\hbox{Br}(F)\) is mentioned but not proved. In Chapter VI on quadratic forms over local and global fields, a subsection on the classification of Witt rings of nonreal fields with four square classes is added. Chapter VII deals with quadratic forms under algebraic extensions. Here, several results have been added on the behaviour of the order of the Witt ring resp. the square class group under quadratic or finite extensions, as well as the Gross-Fischer theorem on obtaining field extensions with prescribed square class group and various results on trace forms. What was previously an appendix to Chapter VII in ATQF has now been considerably extended and become a section on quadratic closures.

Chapter VIII on formally real fields, real-closed fields and Pythagorean fields has undergone substantial changes. Most notably, Euclidean fields are introduced, a proof of the uniqueness of the real closure for an ordered field and a different proof of Pfister’s local-global principle are given as well as Artin-Schreier’s result on subfields of finite index inside an algebraically closed field. Sections on applications to Galois theory, on the Harrison topology, and on preorderings have been added. Chapter IX on quadratic forms under transcendental extensions is more or less unchanged. It contains the results by Cassels-Pfister and the exact sequences concerning the Witt ring of the rational function field in one variable by Milnor and by Scharlau.

In ATQF, function fields of quadratic forms have only shown up very sporadically and implicitly. However, many of the most important recent results in the algebraic theory of quadratic forms concern such function fields. Chapter X, which originally only developed the theory of (strongly) multiplicative and Pfister forms and which, as an application, gave a proof of the Arason-Pfister Hauptsatz, now contains extensive sections on function fields of quadratic forms, addressing among other things the question of isotropy and hyperbolicity of quadratic forms over such function fields, and stating (without proof) some of the recent results by the reviewer, Karpenko, Vishik, Karpenko-Merkurjev and others on questions such as the first higher Witt index, dimensions in \(I^nF\), essential dimension of quadratic forms etc. The chapter also includes a section on the Milnor conjecture.

Chapter XI is concerned with field invariants such as the level, the Pythagoras number and the \(u\)-invariant. This chapter has also undergone substantial changes and now includes a section on the property \(A_n\) and a proof of Leep’s theorem on the possible growth of the \(u\)-invariant under finite extensions.

The last two chapters XII and XIII on special topics in quadratic forms and on invariants are completely new and comprise more than one hundred pages. Without claiming to be a systematic account of specific aspects of the theory, they give however a nice flavor of the various directions in which the theory branches out and of its multifarious facets. Topics include Harrison’s isomorphism criterion for Witt rings, a study of quadratic forms of small dimension, classification of quadratic forms over certain fields, Kaplansky radical, (pre-)Hilbert fields, axiomatic quadratic form theories, the construction of fields with prescribed invariants, the level and Pythagoras number of commutative rings.

At the end of ATQF, Lam listed eight open problems on which much of the research in quadratic forms was focussed at the time. And again, in the present book, Lam closes by posing this time ten open questions that attract current interest or might stimulate further research. These questions pertain mainly to field invariants and their possible values and their behaviour when passing to a purely transcendental extension of a field for which the values are known. One question concerns the isomorphism of function fields of quadratic forms (the “quadratic” Zariski problem). All these questions have deceptively simple formulations, and five of them have already appeared in ATQF (albeit formulated sometimes a little differently due to partial progress toward their solution). It is perhaps an indication of the difficulty of the subject that answers to these questions still elude us more than thirty years after they were asked in ATQF. On the other hand, the many spectacular advances which have taken place over the last thirty years, including solutions to some of the questions asked in ATQF, show that the algebraic theory of quadratic forms over fields has been and continues to be a thriving theory. In fact, the quest for solving some of these questions has necessitated and inspired new developments and the creation of new methods in other areas such as algebraic and arithmetic geometry, Galois cohomology, algebraic \(K\)-theory, central simple algebras, or real algebraic geometry, to name but a few.

The present book is a wonderful achievement. Its genesis is recounted with charm and warmth in the preface. The author’s lucid style and expository skill, his judicious choice of topics and their impeccable layout, not to mention the beautiful typesetting (an obvious improvement compared to the typography of ATQF which was written on an IBM Selectric typewriter) make the book a joy to read or just to browse. Anybody owning a copy of ATQF (most likely worn out and on the verge of disintegrating like the reviewer’s) will certainly want to have a copy of the present book on his or her shelf. But it is sure to attract many new admirers and it will be a must for anybody working in quadratic forms or on topics related to or using quadratic forms, be it for learning the theory of quadratic forms over fields from its foundations, or be it as a reference.

Reviewer: Detlev Hoffmann (Nottingham)

### MSC:

11Exx | Forms and linear algebraic groups |

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

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

15A63 | Quadratic and bilinear forms, inner products |

15A66 | Clifford algebras, spinors |

16K20 | Finite-dimensional division rings |

16K50 | Brauer groups (algebraic aspects) |

16W50 | Graded rings and modules (associative rings and algebras) |