##
**Introduction to quadratic forms.**
*(English)*
Zbl 0107.03301

Die Grundlehren der mathematischen Wissenschaften. 117. Berlin-GĂ¶ttingen-Heidelberg: Springer-Verlag. xi, 342 pp. with 10 fig. (1963).

This is a very readable introduction to the Theory of Quadratic Forms in an up-to-date but not ultramodern setting. (Ideles and adeles – here called split rotations play an important part, but they are not endowed with a topology. There is no homological algebra and no appeal to the duality theory of locally compact groups). As the author stresses in his preface, he has not attempted any complete coverage of the theory of quadratic forms. He is primarily concerned with the relation between “local” and “global” properties and with the arithmetical properties of the orthogonal groups. His point of view is thus rather similar to that of M. Eichler [Quadratische Formen and orthogonale Gruppen. Berlin etc.: Springer-Verlag (1952; Zbl 0049.31106)] but he proceeds at a more leisurely pace, is more lowbrow and aims at a much more modest coverage: and, of course, he can and does take full advantage of the new developments of the last ten years. The book will serve as a useful introduction to the reader interested in contemporary work on quadratic forms and the orthogonal groups.

The author avoids as far as possible mere algebraic manipulation and emphasizes the important concepts and techniques such as: (1) the use of normed vector spaces (2) Witt’s Lemma, (3) the interaction between the arithmetic of a quadratic form and that of its orthogonal group, (4) the role of the spinor norm, (5) the special role of maximal lattices, and (6) the part played by the theorems about the simultaneous approximation of local objects by global ones. The author assumes only the most basic mathematical equipment from his readers and about a third of the book is devoted to algebraic and number-theoretic background material. In particular, there is a proof from scratch of the Hasse Norm Theorem for quadratic extensions of algebraic number fields using the idele group and “counting”. Apart from its relevance to the rest of the book, this part should be useful to a beginner about to embark on the jungle of modern literature on class field theory. The treatment is self-contained. In particular, an appeal to the theorem about the existence of prime ideals in arithmetic progressions, which has hither to apparently been unavoidable (e. g. Eichler, loc. cit., p. 159) is evaded by a simple and elegant device (p. 187). There is, alas, only an inadequate list of notation and, as some of it is not standard, the reviewer was compelled to make a list of some 15 items for his own convenience. This is, of course, a minor matter if you wish to read the book from cover to cover but it might seriously diminish its usefulness as a work of reference. Otherwise, the presentation seems perfectly clear.

The following annotated list of contents gives a more precise notion of the scope of the book: Part One: Arithmetic theory of fields. Ch. I. Valuated fields. Ch. II. Dedekind theory of ideals. Ch. III. Fields of number theory. Part Two: Abstract theory of quadratic forms. Ch. IV. Quadratic forms and the orthogonal group. Ch. V. The algebras of quadratic forms (Clifford algebra, quaternion algebras). Part Three: Arithmetic theory of quadratic forms over fields. Ch. VI. The equivalence of quadratic forms (including that everywhere locally equivalent forms are globally equivalent). Ch. VII. Hilbert’s reciprocity law (includes the condition for the existence of a global form with given localizations). Part Four: Arithmetic theory of quadratic forms over rings. Ch. VIII. Quadratic forms over Dedekind domains. Ch. IX. Integral theory of quadratic forms over local fields. (This chapter classifies quadratic forms over the rings of integers of local fields. In addition to the simple case when 2 is a unit there is an elaborate discussion of the case when it is not – the “dyadic case”. This is probably as well done as is possible in the nature of things but it confirmed the reviewer in his philosophy that the dyadic case is essentially ugly and trivially complicated and to be avoided like the plague. As the author has also adhered to this philosophy in the last chapter of the book by always treating the prime divisors of 2 as exceptional it is a pity that he did not indicate to the less hardy reader that this section could be omitted.) Ch. X. Integral theory of quadratic forms. [This chapter includes a proof that spinor genera and classes coincide for indefinite forms in 3 or more variables [M. Eichler, Math. Z. 55, 216–252 (1952; Zbl 0049.31201); M. Kneser, Arch. Math. 7, 323–332 (1956; Zbl 0071.27205)]. The proof is done not merely when the ground ring is the ring of integers of an algebraic number field, but when it is the ring of field elements integral except at a finite set \(T\) of prime spots (including the infinite primes) and when indefinite means isotropic at a spot of \(T\). As a bonus the author can thus give M. Kneser’s elegant investigation of the class numbers of definite forms by means of the theory of indefinite forms [Arch. Math. 8, 241–250 (1957; Zbl 0078.03801)]. This chapter also includes the theorem that a definite quadratic form can be uniquely decomposed into the sum of indecomposables [M. Eichler, Math. Ann. 125, 51–55 (1952; Zbl 0047.03101); M. Kneser, Math. Ann. 127, 105–106 (1954; Zbl 0055.04304)].

The author avoids as far as possible mere algebraic manipulation and emphasizes the important concepts and techniques such as: (1) the use of normed vector spaces (2) Witt’s Lemma, (3) the interaction between the arithmetic of a quadratic form and that of its orthogonal group, (4) the role of the spinor norm, (5) the special role of maximal lattices, and (6) the part played by the theorems about the simultaneous approximation of local objects by global ones. The author assumes only the most basic mathematical equipment from his readers and about a third of the book is devoted to algebraic and number-theoretic background material. In particular, there is a proof from scratch of the Hasse Norm Theorem for quadratic extensions of algebraic number fields using the idele group and “counting”. Apart from its relevance to the rest of the book, this part should be useful to a beginner about to embark on the jungle of modern literature on class field theory. The treatment is self-contained. In particular, an appeal to the theorem about the existence of prime ideals in arithmetic progressions, which has hither to apparently been unavoidable (e. g. Eichler, loc. cit., p. 159) is evaded by a simple and elegant device (p. 187). There is, alas, only an inadequate list of notation and, as some of it is not standard, the reviewer was compelled to make a list of some 15 items for his own convenience. This is, of course, a minor matter if you wish to read the book from cover to cover but it might seriously diminish its usefulness as a work of reference. Otherwise, the presentation seems perfectly clear.

The following annotated list of contents gives a more precise notion of the scope of the book: Part One: Arithmetic theory of fields. Ch. I. Valuated fields. Ch. II. Dedekind theory of ideals. Ch. III. Fields of number theory. Part Two: Abstract theory of quadratic forms. Ch. IV. Quadratic forms and the orthogonal group. Ch. V. The algebras of quadratic forms (Clifford algebra, quaternion algebras). Part Three: Arithmetic theory of quadratic forms over fields. Ch. VI. The equivalence of quadratic forms (including that everywhere locally equivalent forms are globally equivalent). Ch. VII. Hilbert’s reciprocity law (includes the condition for the existence of a global form with given localizations). Part Four: Arithmetic theory of quadratic forms over rings. Ch. VIII. Quadratic forms over Dedekind domains. Ch. IX. Integral theory of quadratic forms over local fields. (This chapter classifies quadratic forms over the rings of integers of local fields. In addition to the simple case when 2 is a unit there is an elaborate discussion of the case when it is not – the “dyadic case”. This is probably as well done as is possible in the nature of things but it confirmed the reviewer in his philosophy that the dyadic case is essentially ugly and trivially complicated and to be avoided like the plague. As the author has also adhered to this philosophy in the last chapter of the book by always treating the prime divisors of 2 as exceptional it is a pity that he did not indicate to the less hardy reader that this section could be omitted.) Ch. X. Integral theory of quadratic forms. [This chapter includes a proof that spinor genera and classes coincide for indefinite forms in 3 or more variables [M. Eichler, Math. Z. 55, 216–252 (1952; Zbl 0049.31201); M. Kneser, Arch. Math. 7, 323–332 (1956; Zbl 0071.27205)]. The proof is done not merely when the ground ring is the ring of integers of an algebraic number field, but when it is the ring of field elements integral except at a finite set \(T\) of prime spots (including the infinite primes) and when indefinite means isotropic at a spot of \(T\). As a bonus the author can thus give M. Kneser’s elegant investigation of the class numbers of definite forms by means of the theory of indefinite forms [Arch. Math. 8, 241–250 (1957; Zbl 0078.03801)]. This chapter also includes the theorem that a definite quadratic form can be uniquely decomposed into the sum of indecomposables [M. Eichler, Math. Ann. 125, 51–55 (1952; Zbl 0047.03101); M. Kneser, Math. Ann. 127, 105–106 (1954; Zbl 0055.04304)].

Reviewer: J. W. S. Cassels

### MSC:

11Exx | Forms and linear algebraic groups |

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