##
**Quadratic and Hermitian forms.**
*(English)*
Zbl 0584.10010

Grundlehren der Mathematischen Wissenschaften, 270. Berlin etc.: Springer-Verlag. x, 421 p. DM 138.00 (1985).

\(\{\)Editorial remark: The following is an almost literal translation of the original review (in German) in Jahresber. DMV to make it accessible to a broader readership.\(\}\)

This monograph treats the algebraic theory of quadratic and Hermitian forms as a separate and methodologically independent area of modern algebra. The author is a well-known expert in this field who has contributed substantially to the progress derived within the last 20 years. The monograph itself has matured gradually during the last 10 years. This explains why such an enormous wealth of material can be found within 400 pages and this becomes clear when comparing it with the existing well-known monographs in this field, especially O. T. O’Meara [Introduction to quadratic forms (1963; Zbl 0107.033), 3rd ed. (1973; Zbl 0259.10018)], T. Y. Lam [The algebraic theory of quadratic forms (1973; Zbl 0259.10019), 2nd ed. (1980; Zbl 0437.10006)], J. Milnor and D. Husemoller [Symmetric bilinear forms (1973; Zbl 0292.10016)] and J. W. S. Cassels [Rational quadratic forms (1978; Zbl 0395.10029)].

A large amount of the contents of the cited monographs can be found also in the book under review, but moreover it contains quadratic and Hermitian forms in additive categories, involutions in simple algebras and the classification of Hermitian forms over number fields for the first time within a monograph. From this follows that it is written in a relatively concise way and at some places the beginner will have some difficulties in reading it. Advanced students, however, should be able to meet the demands and to follow the proofs. Only very few results, which for good reasons do not belong to the scope of this monograph, are cited from the literature without giving proofs. This concerns especially theorems from algebraic number theory and class field theory as well as Hilbert’s Nullstellensatz.

Apart from that the book is self-contained, some very important theorems are proved several times, and the chapters are organized in such a way that they can be read largely independently. In conclusion: An excellent book, ideally suiting the series title “Grundlehren der Mathematischen Wissenschaften”.

The contents is as follows: Chapter Basic concepts” contains the fundamentals on symmetric bilinear forms and quadratic forms (and spaces) up to and including Witt’s theorem, all over a field K of characteristic different from 2. In an appendix (§6) the concept is generalized to the case of a commutative ring \(R\) with \(2\in R^*\) instead of \(K\) and some theorems are proved when in addition \(R\) is local.

Chapter 2 “Quadratic Forms over Fields” is the largest chapter covering nearly 80 pages. Here one can find the Witt ring \(W(K)\) and Grothendieck-Witt ring \(\hat W(K)\) of symmetric bilinear forms over \(K\), the invariants such as dimension, discriminant and signature with respect to an ordering of \(K\), the computation of \(W(K)\) and \(\hat W(K)\) for finite fields and Euclidean fields (e.g. \(K=\mathbb{R})\), and the theorem that \(K\) is pythagorean if and only if \(\hat W(K)\) is torsionfree.

Further for a field extension \(r_{L/K}\colon K\to L\) the induced map \(r^*_{L/K}\colon W(K)\to W(L)\) and (in case \(L/K\) is finite) for a \(K\)-linear map \(s\colon L\to K\) the induced “Scharlau trace” \(s_*\colon W(L)\to W(K)\) are treated. Herewith are studied especially the cases \([L:K]=2m+1\) (Theorem of T. A. Springer), \([L:K]=2\) (exact triangle of Elman-Lam), and the torsion subgroup \(W_t(K)\) of \(W(K)\) (only 2-torsion exists).

Thereafter follow Pfister’s local-global principle \((\Phi \in W_t(K)\Leftrightarrow \text{sign}_P(\Phi)=0\) for all orderings \(P\) of \(K\), the determination of the prime ideals and the radical of \(W(K)\), the application of the transfer method to the case of a Galois extension (Rosenberg-Ware theorem), the description of the Witt ring by generators and relations (Witt), and its generalization to suitable group rings (Knebusch-Rosenberg-Ware). Then multiplicative forms are introduced and used to prove anew some structure theorems for \(W(K)\).

After a short paragraph of quaternion algebras the treatment of Hasse-(Witt-) invariants follows using Steinberg symbols and Milnor’s theorem that \((\alpha,\beta)\to \langle 1,-\alpha,-\beta,\alpha \beta \rangle\) induces a universal symbol \(\sigma\colon \dot K\times \dot K\to I^2/I^3\) \((I\) the fundamental ideal of forms of even dimension in \(W(K)\)).

Merkurjev’s famous theorem (1982), \(I^ 2/I^ 3\cong \text{Br}_2(K)\) (subgroup of elements of order \(\leq 2\) in the Brauer group of \(K\)) is mentioned but not proved. No mention is made of Milnor’s conjectures on the relation of higher factor groups \(I^n/I^{n+1}\) with corresponding groups of algebraic K-theory and cohomology theory.

However the classification theorem of Elman-Lam is completely proved, by which quadratic forms over \(K\) are classified by the “classical” invariants dimension, discriminant, Witt invariant and all signatures if and only if \(I^3(K)\) is torsionfree.

The two final paragraphs of this chapter show that, besides the local and global fields treated in later chapters, there exists a large class of fields for which the theory developed leads to concrete results on quadratic forms. §15 treats \(C_i\)-fields and contains, for instance, a super short proof of the Chevalley-Warning theorem on finite fields, an elegant proof (after Nastold) of Hilbert’s homogeneous Nullstellensatz from the ordinary Nullstellensatz, and the main theorems on \(C_ i\)-fields (Tsen, Lang, Nagata) with proofs.

§16 studies the \(u\)-invariant of non-real fields (i.e. we have always \(u\neq 3,5,7)\), a theorem of M. Kneser on the values of quadratic forms (from which \(u\leq \vert \dot K/\dot K^2\vert\) follows) and the theorem of Leep (1983) on systems of quadratic forms.

Chapter 3 “Quadratic Forms over Formally Real Fields” starts with Artin-Schreier theory of (formally) real and real-closed fields, contains an elegant proof (after F. Lorenz) of S. Lang’s homomorphy theorem, Artin’s solution of Hilbert’s 17th Problem on positive definite rational functions over real varieties, the real Nullstellensatz of Dubois-Risler with a proof of Prestel and theorems on the extension of orderings and signatures to finite field extensions \(L/K\).

One can also find here the space \(X_ K\) of all orderings \(P\) of the real field \(K\) with its most important algebraic and topological properties, the total signature \(\wedge\colon W(K)\to C(X_K,\mathbb{Z})\), the kernel \(W_ t\) and cokernel of which are 2-torsion groups, respectively (this proves anew the local-global principle for \(W(K)\), and finally a proof of Bröcker and Prestel of the important local-global principle for weak isotropy. For this one has to develop the theory of quadratic semiorders, and an appendix presents some fundamentals on valuations and their relations with orderings. A form is called weakly isotropic if the form \(m\times \Phi =\Phi +\dots+\Phi\) is isotropic for a sufficiently large positive integer \(m\).

Chapter 4 “Generic Methods and Pfister Forms” contains all those results on quadratic forms, in which transcendental extensions play a substantial role. To them belong the representation theorems of Cassels, the characterization of the forms \(\langle\langle a_1,\dots, a_n\rangle\rangle =\otimes^n_{i=1}\langle 1,a_i\rangle\) as generic multiplicative forms, Krull’s intersection theorem \(\cap_{n}I^n=0\) and, necessary for its proof, the “Main Theorem of Arason and Pfister” (in the terminology of Lam), and introduced by Knebusch the generic zero and splitting fields of an anisotropic form as well as the filtration of the Witt ring by the ideals \(J_n=\{\phi \in W\mid \deg\phi\geq n\}\) of Arason-Knebusch where \(\deg\phi\) \(=d\), if the “leading” form \(\rho\) with respect to \(\phi\) has dimension \(2^d\); \(I^n\subset J_n\) is valid and the question of equality is one of the more difficult still unsolved problems. Applications of these generic methods, as e.g. the quantitative solution of Hilbert’s 17th problem for the field \(\mathbb{R}(X_1,\dots, X_n)\) and the construction of fields with given level \(s=2^k\) are fitted in when suitable.

Chapter 5 ”Rational Quadratic Forms” is very remarkable. It treats the theory of Gauss and Minkowski but in a very modern and original style the fruitfulness of which shows up later on in chapters 6 and 7.

The chapter begins with symmetric bilinear forms and quadratic forms over finite abelian groups \(M\) with values in \(\mathbb{Q}/\mathbb{Z}\). For special \(M\) they lead to a Witt ring \(W\) and a Witt group \(WQ\). These are computed using Gauss sums, and for the 2-primary part of \(WQ\) we have e.g. \[ WQ(2)=\mathbb{Z}/8\mathbb{Z}+\mathbb{Z}/2\mathbb{Z}.\]

Then follows the theory of \(\mathbb{Z}\)-lattices \(L\). If \(L^{\ne}\) is the dual lattice, then \(M=L^{\ne}/L\) is a finite abelian group of order \(| \det L|\). After introducing the (second) residue class homomorphisms \(\partial_p\colon W(\mathbb{Q})\to W(\mathbb{F}_p)\) it is not difficult to compute completely and explicitly the Witt groups \(W(\mathbb{Q})\) and \(W(\mathbb{Z})\). It follows \[ W(\mathbb{Q})\cong W(\mathbb{R})\oplus W\cong \mathbb{Z}\oplus \oplus_p W(\mathbb{F}_p),\quad W(\mathbb{Z})\cong \mathbb{Z}. \] The proofs were given by Milnor.

The following sections contain a proof of the quadratic reciprocity law using a canonical homomorphism \(\chi\colon W(\mathbb{Q})\to T\) into the group \(T\) of eighth roots of unity, the complete classification of quadratic forms over the \(p\)-adic field \(\mathbb{Q}_p\), Hasse-Minkowski’s theorem for \(\mathbb{Q}\) (for its proof Dirichlet’s prime number theorem is used) and the explicit calculation of the Gaussian sums \(G(2m)=\sum^{2m}_{k=1}e^{(\pi ik^2/2m)}.\)

Chapter 6 “Symmetric Bilinear Forms over Dedekind Rings and Global fields” generalizes the results of Chapter 5 to Dedekind rings \(R\) and their quotient fields \(K\) (under the restriction \(\text{char}\,R\neq 2)\). Some fundamentals on Dedekind rings are assumed or cited. The central theme are the following exact sequences and isomorphisms. Let \(WT(R)\) be the Witt group of regular symmetric bilinear forms over finitely generated \(R\)-torsion modules, identifying weak metabolic forms with zero. Then the canonical sequence

\[ 0\rightarrow W(R)\rightarrow W(K)\overset{\delta}\rightarrow WT(R) \]

is exact and \(W(T)\cong \coprod_p WT(R_{\mathfrak p})\) (Milnor) and \(WT(R_{\mathfrak p})\cong W(k_{\mathfrak p})\) (depending on the choice of the prime element \(\pi\) in \(R_{\mathfrak p})\). If \(R_{\mathfrak p}\) is 2-henselian and \(\text{char}\,k_{\mathfrak p}\neq 2\), then also \(W(R_{\mathfrak p})\cong W(k_p)\), and therefore \(W(K_{\mathfrak p})=W(k_{\mathfrak p})+W(k_{\mathfrak p})\) (Springer). Thereby quadratic forms over \(p\)-adic fields can be treated completely (for some details in case of residue class characteristic 2 one is referred to O’Meara). If \(R=k[X]\) is a polynomial ring, then the following sharper results are valid: \(\delta\) is surjective (Milnor), \(W(R)\cong W(k)\) (Harder),

\[ W(K)\overset{\delta}\rightarrow\coprod_{\mathfrak p,\infty}W(k_{\mathfrak p})\overset{s}\rightarrow W(k)\to 0 \]

is exact (Scharlau). If \(K\) is a global field, then first the theorem of Hasse-Minkowski is proved with all its conclusions and supplements (existence of forms with prescribed local invariants etc.), however giving only citations for the global square theorem and Hasse’s norm theorem. Then the exactness of the sequence

\[ 0\rightarrow W(R)\to W(K)\overset{\delta}\rightarrow WT(R)\to C/C^2\to 0\quad \text{(Knebusch-Scharlau)} \]

is proved where \(C\) is the ideal class group of \(R\).

Finally the chapter contains a nice proof of Hilbert’s reciprocity law for global fields \(K\) by reduction to the “rational” subfield \(\mathbb{Q}\) or \(k(X)\), respectively, a proof of Hecke’s theorem on the different and the residue theorem of Geyer-Harder-Knebusch-Scharlau for differential-valued quadratic forms in algebraic function fields.

Chapter 7 “Foundations of the Theory of Hermitian Forms” has two aims, on one hand the introduction of Hermitian forms, on the other the foundation of a general theory of forms in additive categories. The main objects are a ring with involution \({}^*\), an element \(\lambda\) of the center of \(R\) with \(\lambda \lambda^*=1\), \(\lambda\)-Hermitian \(R\)-modules \((M,h)\) with \(h(x,y)=\lambda h(y,x)^*,\) the duality \(h\colon M\to M^*= \operatorname{Hom}(M,R),\) hyperbolic spaces, the Grothendieck group \(KU^{\lambda}(R,^*)\) and the Witt group \(W^{\lambda}(R,^*)\). To include forms over torsion modules one defines more generally so-called Hermitian categories (\(\mathcal M,^*)\). Next, according to Bak, form parameters \(\Lambda\), quadratic forms and finally the Witt group \(WQ^{\lambda,\Lambda}(\mathcal M,^*)\) are defined. In the case \((R,^*)\) above a form parameter \(\Lambda\) is an additive subgroup of \(R\) with \[ \{a-\lambda a^*\mid a\in R\}=\Lambda^{\min}\subset \Lambda \subset \Lambda^{\max}=\{a\in R \mid a=-\lambda a^*\}, \] the quadratic form with respect to \(h\) is given by \(q_h\colon M\to R/\Lambda\) and \(q_h(x)=h(x,x)+\Lambda\). Subsequently the following matters are treated under suitable conditions: transfer and reduction of Hermitian categories, Hermitian forms over skew fields, hyperbolic forms and the unitary group, alternating forms and the symplectic group, quadratic forms and the orthogonal group (Theorem of Witt and Cartan-Dieudonné under very general conditions) and the Krull-Schmidt theorem (direct decomposition into indecomposables).

Some examples (a.o. systems of quadratic forms over a field \(k\)) show the benefit of this general and rather technical theory.

Chapter 8 “Simple Algebras and Involutions” consists of the classical theory of simple algebras including the theorems of Wedderburn and Skolem-Noether and the Brauer group. Next follows Albert’s theory on the existence and classification of involutions on simple algebras in an elegant version after Scharlau and Riehm. There is mentioned the reduced norm and trace as well as the corestriction of algebras or separable quadratic extensions. Finally quaternion algebras (also in characteristic 2), cyclic algebras and the canonical involution on a group algebra \(A=K[G]\) are treated.

Chapter 9 “Clifford Algebras” collects in a short but on the whole classical treatment the most important facts on Clifford algebras and Clifford \((=\) Witt) invariant of a quadratic space, spinor norm and Arf invariant.

Chapter 10 “Hermitian Forms over Global Fields” is devoted to the classification (by local and global invariants) and the local-global principle for \(\lambda\)-Hermitian forms over global fields of characteristic \(\neq 2\). Most of the results go back to Landherr (1938), Eichler and M. Kneser. They are derived in a unified manner using results from chapters 7 and 8 and cited results from algebraic number theory. Nevertheless the chapter covers more than 40 pages. According to the nature of the base field and the involution the following cases occur:

1) Orthogonal case: Quadratic forms over fields (already treated) and skew-Hermitian (i.e. \(\lambda =-1)\) forms over quaternion skew-fields. (§§3 and 4).

2) Symplectic case: Alternating forms over fields (trivial) and Hermitian forms (i.e. \(\lambda =1)\) over quaternion skew-fields (§1).

3) Unitary case: Hermitian forms over fields with nontrivial involution \({}^*\) (§1) and Hermitian forms over skew-fields \(D\) with unitary involution \({}^*\) (i.e. \({}^*\) is nontrivial on the center \(K\) of \(D\); here for \([D:K]=n^2\) the case \(n>2\) can occur). This most difficult case is studied in §§6–8.

The most general case can be reduced to the said one, however, taking into account that a skew field \(D\) can also split when going over to a localization \(K_{\mathfrak p}\) of \(K\). In §1 for a separable quadratic extension \(K/k\) with nontrivial automorphism \({}^*\) the exact sequence \[ 0\to W^1(K,^*)\to W(k)\to W(K) \] is derived. Moreover it is shown that for a quaternion skew field \(D\) with canonical involution \({}^*\) and center \(K\) the natural map \(h\to q_h\) where \(q_h(x)=h(x,x)\) induces an injection \(W^1(D,^*)\to W(K)\). This reduces both cases to quadratic forms.

§2 collects the necessary theorems on Brauer groups of local and global fields. These are used to treat in §3 the local and in §4 the global theory of skew-Hermitian forms over quaternion skew-fields \(D/K\) with intermediate field \(L/K\). Here the following exact sequence is fundamental: \[ 0\to W^ 1(D,^*)\to W^ 1(L,^*)\to W^{-1}(D,^*)\to W(L). \] Here the strong local-global principle for isotropy is only correct for \(\dim h\geq 3.\)

§5 gives a special case of the strong approximation theorem for the special linear group (of a global field \(K\)) including a sketch of proof. This is used in §§6–8 to settle the unitary case over skew-fields \(D\). Here the strong local-global-principle is valid without any restrictions.

Four short appendices conclude this substantial book. They are concerned with the following “pearls” rounding off the actual theory: the level of commutative rings, quadratic forms on finite graphs and their relations to Dynkin diagrams \(A_n\), \(D_n\), \(E_6\), \(E_7\), \(E_8\) and quivers of finite representation type (Gabriel); the five platonic polyhedra; the Hermitian inequality and the positive definite even unimodular symmetric bilinear form over \(\mathbb{Z}\) of dimension 8.

This monograph treats the algebraic theory of quadratic and Hermitian forms as a separate and methodologically independent area of modern algebra. The author is a well-known expert in this field who has contributed substantially to the progress derived within the last 20 years. The monograph itself has matured gradually during the last 10 years. This explains why such an enormous wealth of material can be found within 400 pages and this becomes clear when comparing it with the existing well-known monographs in this field, especially O. T. O’Meara [Introduction to quadratic forms (1963; Zbl 0107.033), 3rd ed. (1973; Zbl 0259.10018)], T. Y. Lam [The algebraic theory of quadratic forms (1973; Zbl 0259.10019), 2nd ed. (1980; Zbl 0437.10006)], J. Milnor and D. Husemoller [Symmetric bilinear forms (1973; Zbl 0292.10016)] and J. W. S. Cassels [Rational quadratic forms (1978; Zbl 0395.10029)].

A large amount of the contents of the cited monographs can be found also in the book under review, but moreover it contains quadratic and Hermitian forms in additive categories, involutions in simple algebras and the classification of Hermitian forms over number fields for the first time within a monograph. From this follows that it is written in a relatively concise way and at some places the beginner will have some difficulties in reading it. Advanced students, however, should be able to meet the demands and to follow the proofs. Only very few results, which for good reasons do not belong to the scope of this monograph, are cited from the literature without giving proofs. This concerns especially theorems from algebraic number theory and class field theory as well as Hilbert’s Nullstellensatz.

Apart from that the book is self-contained, some very important theorems are proved several times, and the chapters are organized in such a way that they can be read largely independently. In conclusion: An excellent book, ideally suiting the series title “Grundlehren der Mathematischen Wissenschaften”.

The contents is as follows: Chapter Basic concepts” contains the fundamentals on symmetric bilinear forms and quadratic forms (and spaces) up to and including Witt’s theorem, all over a field K of characteristic different from 2. In an appendix (§6) the concept is generalized to the case of a commutative ring \(R\) with \(2\in R^*\) instead of \(K\) and some theorems are proved when in addition \(R\) is local.

Chapter 2 “Quadratic Forms over Fields” is the largest chapter covering nearly 80 pages. Here one can find the Witt ring \(W(K)\) and Grothendieck-Witt ring \(\hat W(K)\) of symmetric bilinear forms over \(K\), the invariants such as dimension, discriminant and signature with respect to an ordering of \(K\), the computation of \(W(K)\) and \(\hat W(K)\) for finite fields and Euclidean fields (e.g. \(K=\mathbb{R})\), and the theorem that \(K\) is pythagorean if and only if \(\hat W(K)\) is torsionfree.

Further for a field extension \(r_{L/K}\colon K\to L\) the induced map \(r^*_{L/K}\colon W(K)\to W(L)\) and (in case \(L/K\) is finite) for a \(K\)-linear map \(s\colon L\to K\) the induced “Scharlau trace” \(s_*\colon W(L)\to W(K)\) are treated. Herewith are studied especially the cases \([L:K]=2m+1\) (Theorem of T. A. Springer), \([L:K]=2\) (exact triangle of Elman-Lam), and the torsion subgroup \(W_t(K)\) of \(W(K)\) (only 2-torsion exists).

Thereafter follow Pfister’s local-global principle \((\Phi \in W_t(K)\Leftrightarrow \text{sign}_P(\Phi)=0\) for all orderings \(P\) of \(K\), the determination of the prime ideals and the radical of \(W(K)\), the application of the transfer method to the case of a Galois extension (Rosenberg-Ware theorem), the description of the Witt ring by generators and relations (Witt), and its generalization to suitable group rings (Knebusch-Rosenberg-Ware). Then multiplicative forms are introduced and used to prove anew some structure theorems for \(W(K)\).

After a short paragraph of quaternion algebras the treatment of Hasse-(Witt-) invariants follows using Steinberg symbols and Milnor’s theorem that \((\alpha,\beta)\to \langle 1,-\alpha,-\beta,\alpha \beta \rangle\) induces a universal symbol \(\sigma\colon \dot K\times \dot K\to I^2/I^3\) \((I\) the fundamental ideal of forms of even dimension in \(W(K)\)).

Merkurjev’s famous theorem (1982), \(I^ 2/I^ 3\cong \text{Br}_2(K)\) (subgroup of elements of order \(\leq 2\) in the Brauer group of \(K\)) is mentioned but not proved. No mention is made of Milnor’s conjectures on the relation of higher factor groups \(I^n/I^{n+1}\) with corresponding groups of algebraic K-theory and cohomology theory.

However the classification theorem of Elman-Lam is completely proved, by which quadratic forms over \(K\) are classified by the “classical” invariants dimension, discriminant, Witt invariant and all signatures if and only if \(I^3(K)\) is torsionfree.

The two final paragraphs of this chapter show that, besides the local and global fields treated in later chapters, there exists a large class of fields for which the theory developed leads to concrete results on quadratic forms. §15 treats \(C_i\)-fields and contains, for instance, a super short proof of the Chevalley-Warning theorem on finite fields, an elegant proof (after Nastold) of Hilbert’s homogeneous Nullstellensatz from the ordinary Nullstellensatz, and the main theorems on \(C_ i\)-fields (Tsen, Lang, Nagata) with proofs.

§16 studies the \(u\)-invariant of non-real fields (i.e. we have always \(u\neq 3,5,7)\), a theorem of M. Kneser on the values of quadratic forms (from which \(u\leq \vert \dot K/\dot K^2\vert\) follows) and the theorem of Leep (1983) on systems of quadratic forms.

Chapter 3 “Quadratic Forms over Formally Real Fields” starts with Artin-Schreier theory of (formally) real and real-closed fields, contains an elegant proof (after F. Lorenz) of S. Lang’s homomorphy theorem, Artin’s solution of Hilbert’s 17th Problem on positive definite rational functions over real varieties, the real Nullstellensatz of Dubois-Risler with a proof of Prestel and theorems on the extension of orderings and signatures to finite field extensions \(L/K\).

One can also find here the space \(X_ K\) of all orderings \(P\) of the real field \(K\) with its most important algebraic and topological properties, the total signature \(\wedge\colon W(K)\to C(X_K,\mathbb{Z})\), the kernel \(W_ t\) and cokernel of which are 2-torsion groups, respectively (this proves anew the local-global principle for \(W(K)\), and finally a proof of Bröcker and Prestel of the important local-global principle for weak isotropy. For this one has to develop the theory of quadratic semiorders, and an appendix presents some fundamentals on valuations and their relations with orderings. A form is called weakly isotropic if the form \(m\times \Phi =\Phi +\dots+\Phi\) is isotropic for a sufficiently large positive integer \(m\).

Chapter 4 “Generic Methods and Pfister Forms” contains all those results on quadratic forms, in which transcendental extensions play a substantial role. To them belong the representation theorems of Cassels, the characterization of the forms \(\langle\langle a_1,\dots, a_n\rangle\rangle =\otimes^n_{i=1}\langle 1,a_i\rangle\) as generic multiplicative forms, Krull’s intersection theorem \(\cap_{n}I^n=0\) and, necessary for its proof, the “Main Theorem of Arason and Pfister” (in the terminology of Lam), and introduced by Knebusch the generic zero and splitting fields of an anisotropic form as well as the filtration of the Witt ring by the ideals \(J_n=\{\phi \in W\mid \deg\phi\geq n\}\) of Arason-Knebusch where \(\deg\phi\) \(=d\), if the “leading” form \(\rho\) with respect to \(\phi\) has dimension \(2^d\); \(I^n\subset J_n\) is valid and the question of equality is one of the more difficult still unsolved problems. Applications of these generic methods, as e.g. the quantitative solution of Hilbert’s 17th problem for the field \(\mathbb{R}(X_1,\dots, X_n)\) and the construction of fields with given level \(s=2^k\) are fitted in when suitable.

Chapter 5 ”Rational Quadratic Forms” is very remarkable. It treats the theory of Gauss and Minkowski but in a very modern and original style the fruitfulness of which shows up later on in chapters 6 and 7.

The chapter begins with symmetric bilinear forms and quadratic forms over finite abelian groups \(M\) with values in \(\mathbb{Q}/\mathbb{Z}\). For special \(M\) they lead to a Witt ring \(W\) and a Witt group \(WQ\). These are computed using Gauss sums, and for the 2-primary part of \(WQ\) we have e.g. \[ WQ(2)=\mathbb{Z}/8\mathbb{Z}+\mathbb{Z}/2\mathbb{Z}.\]

Then follows the theory of \(\mathbb{Z}\)-lattices \(L\). If \(L^{\ne}\) is the dual lattice, then \(M=L^{\ne}/L\) is a finite abelian group of order \(| \det L|\). After introducing the (second) residue class homomorphisms \(\partial_p\colon W(\mathbb{Q})\to W(\mathbb{F}_p)\) it is not difficult to compute completely and explicitly the Witt groups \(W(\mathbb{Q})\) and \(W(\mathbb{Z})\). It follows \[ W(\mathbb{Q})\cong W(\mathbb{R})\oplus W\cong \mathbb{Z}\oplus \oplus_p W(\mathbb{F}_p),\quad W(\mathbb{Z})\cong \mathbb{Z}. \] The proofs were given by Milnor.

The following sections contain a proof of the quadratic reciprocity law using a canonical homomorphism \(\chi\colon W(\mathbb{Q})\to T\) into the group \(T\) of eighth roots of unity, the complete classification of quadratic forms over the \(p\)-adic field \(\mathbb{Q}_p\), Hasse-Minkowski’s theorem for \(\mathbb{Q}\) (for its proof Dirichlet’s prime number theorem is used) and the explicit calculation of the Gaussian sums \(G(2m)=\sum^{2m}_{k=1}e^{(\pi ik^2/2m)}.\)

Chapter 6 “Symmetric Bilinear Forms over Dedekind Rings and Global fields” generalizes the results of Chapter 5 to Dedekind rings \(R\) and their quotient fields \(K\) (under the restriction \(\text{char}\,R\neq 2)\). Some fundamentals on Dedekind rings are assumed or cited. The central theme are the following exact sequences and isomorphisms. Let \(WT(R)\) be the Witt group of regular symmetric bilinear forms over finitely generated \(R\)-torsion modules, identifying weak metabolic forms with zero. Then the canonical sequence

\[ 0\rightarrow W(R)\rightarrow W(K)\overset{\delta}\rightarrow WT(R) \]

is exact and \(W(T)\cong \coprod_p WT(R_{\mathfrak p})\) (Milnor) and \(WT(R_{\mathfrak p})\cong W(k_{\mathfrak p})\) (depending on the choice of the prime element \(\pi\) in \(R_{\mathfrak p})\). If \(R_{\mathfrak p}\) is 2-henselian and \(\text{char}\,k_{\mathfrak p}\neq 2\), then also \(W(R_{\mathfrak p})\cong W(k_p)\), and therefore \(W(K_{\mathfrak p})=W(k_{\mathfrak p})+W(k_{\mathfrak p})\) (Springer). Thereby quadratic forms over \(p\)-adic fields can be treated completely (for some details in case of residue class characteristic 2 one is referred to O’Meara). If \(R=k[X]\) is a polynomial ring, then the following sharper results are valid: \(\delta\) is surjective (Milnor), \(W(R)\cong W(k)\) (Harder),

\[ W(K)\overset{\delta}\rightarrow\coprod_{\mathfrak p,\infty}W(k_{\mathfrak p})\overset{s}\rightarrow W(k)\to 0 \]

is exact (Scharlau). If \(K\) is a global field, then first the theorem of Hasse-Minkowski is proved with all its conclusions and supplements (existence of forms with prescribed local invariants etc.), however giving only citations for the global square theorem and Hasse’s norm theorem. Then the exactness of the sequence

\[ 0\rightarrow W(R)\to W(K)\overset{\delta}\rightarrow WT(R)\to C/C^2\to 0\quad \text{(Knebusch-Scharlau)} \]

is proved where \(C\) is the ideal class group of \(R\).

Finally the chapter contains a nice proof of Hilbert’s reciprocity law for global fields \(K\) by reduction to the “rational” subfield \(\mathbb{Q}\) or \(k(X)\), respectively, a proof of Hecke’s theorem on the different and the residue theorem of Geyer-Harder-Knebusch-Scharlau for differential-valued quadratic forms in algebraic function fields.

Chapter 7 “Foundations of the Theory of Hermitian Forms” has two aims, on one hand the introduction of Hermitian forms, on the other the foundation of a general theory of forms in additive categories. The main objects are a ring with involution \({}^*\), an element \(\lambda\) of the center of \(R\) with \(\lambda \lambda^*=1\), \(\lambda\)-Hermitian \(R\)-modules \((M,h)\) with \(h(x,y)=\lambda h(y,x)^*,\) the duality \(h\colon M\to M^*= \operatorname{Hom}(M,R),\) hyperbolic spaces, the Grothendieck group \(KU^{\lambda}(R,^*)\) and the Witt group \(W^{\lambda}(R,^*)\). To include forms over torsion modules one defines more generally so-called Hermitian categories (\(\mathcal M,^*)\). Next, according to Bak, form parameters \(\Lambda\), quadratic forms and finally the Witt group \(WQ^{\lambda,\Lambda}(\mathcal M,^*)\) are defined. In the case \((R,^*)\) above a form parameter \(\Lambda\) is an additive subgroup of \(R\) with \[ \{a-\lambda a^*\mid a\in R\}=\Lambda^{\min}\subset \Lambda \subset \Lambda^{\max}=\{a\in R \mid a=-\lambda a^*\}, \] the quadratic form with respect to \(h\) is given by \(q_h\colon M\to R/\Lambda\) and \(q_h(x)=h(x,x)+\Lambda\). Subsequently the following matters are treated under suitable conditions: transfer and reduction of Hermitian categories, Hermitian forms over skew fields, hyperbolic forms and the unitary group, alternating forms and the symplectic group, quadratic forms and the orthogonal group (Theorem of Witt and Cartan-Dieudonné under very general conditions) and the Krull-Schmidt theorem (direct decomposition into indecomposables).

Some examples (a.o. systems of quadratic forms over a field \(k\)) show the benefit of this general and rather technical theory.

Chapter 8 “Simple Algebras and Involutions” consists of the classical theory of simple algebras including the theorems of Wedderburn and Skolem-Noether and the Brauer group. Next follows Albert’s theory on the existence and classification of involutions on simple algebras in an elegant version after Scharlau and Riehm. There is mentioned the reduced norm and trace as well as the corestriction of algebras or separable quadratic extensions. Finally quaternion algebras (also in characteristic 2), cyclic algebras and the canonical involution on a group algebra \(A=K[G]\) are treated.

Chapter 9 “Clifford Algebras” collects in a short but on the whole classical treatment the most important facts on Clifford algebras and Clifford \((=\) Witt) invariant of a quadratic space, spinor norm and Arf invariant.

Chapter 10 “Hermitian Forms over Global Fields” is devoted to the classification (by local and global invariants) and the local-global principle for \(\lambda\)-Hermitian forms over global fields of characteristic \(\neq 2\). Most of the results go back to Landherr (1938), Eichler and M. Kneser. They are derived in a unified manner using results from chapters 7 and 8 and cited results from algebraic number theory. Nevertheless the chapter covers more than 40 pages. According to the nature of the base field and the involution the following cases occur:

1) Orthogonal case: Quadratic forms over fields (already treated) and skew-Hermitian (i.e. \(\lambda =-1)\) forms over quaternion skew-fields. (§§3 and 4).

2) Symplectic case: Alternating forms over fields (trivial) and Hermitian forms (i.e. \(\lambda =1)\) over quaternion skew-fields (§1).

3) Unitary case: Hermitian forms over fields with nontrivial involution \({}^*\) (§1) and Hermitian forms over skew-fields \(D\) with unitary involution \({}^*\) (i.e. \({}^*\) is nontrivial on the center \(K\) of \(D\); here for \([D:K]=n^2\) the case \(n>2\) can occur). This most difficult case is studied in §§6–8.

The most general case can be reduced to the said one, however, taking into account that a skew field \(D\) can also split when going over to a localization \(K_{\mathfrak p}\) of \(K\). In §1 for a separable quadratic extension \(K/k\) with nontrivial automorphism \({}^*\) the exact sequence \[ 0\to W^1(K,^*)\to W(k)\to W(K) \] is derived. Moreover it is shown that for a quaternion skew field \(D\) with canonical involution \({}^*\) and center \(K\) the natural map \(h\to q_h\) where \(q_h(x)=h(x,x)\) induces an injection \(W^1(D,^*)\to W(K)\). This reduces both cases to quadratic forms.

§2 collects the necessary theorems on Brauer groups of local and global fields. These are used to treat in §3 the local and in §4 the global theory of skew-Hermitian forms over quaternion skew-fields \(D/K\) with intermediate field \(L/K\). Here the following exact sequence is fundamental: \[ 0\to W^ 1(D,^*)\to W^ 1(L,^*)\to W^{-1}(D,^*)\to W(L). \] Here the strong local-global principle for isotropy is only correct for \(\dim h\geq 3.\)

§5 gives a special case of the strong approximation theorem for the special linear group (of a global field \(K\)) including a sketch of proof. This is used in §§6–8 to settle the unitary case over skew-fields \(D\). Here the strong local-global-principle is valid without any restrictions.

Four short appendices conclude this substantial book. They are concerned with the following “pearls” rounding off the actual theory: the level of commutative rings, quadratic forms on finite graphs and their relations to Dynkin diagrams \(A_n\), \(D_n\), \(E_6\), \(E_7\), \(E_8\) and quivers of finite representation type (Gabriel); the five platonic polyhedra; the Hermitian inequality and the positive definite even unimodular symmetric bilinear form over \(\mathbb{Z}\) of dimension 8.

Reviewer: Albrecht Pfister (Mainz)

### MSC:

11Exx | Forms and linear algebraic groups |

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

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

12-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory |

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

11E04 | Quadratic forms over general fields |

11E08 | Quadratic forms over local rings and fields |

11E10 | Forms over real fields |

11E12 | Quadratic forms over global rings and fields |

11E16 | General binary quadratic forms |

11E39 | Bilinear and Hermitian forms |

11E57 | Classical groups |

11E88 | Quadratic spaces; Clifford algebras |

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

12E15 | Skew fields, division rings |

15A63 | Quadratic and bilinear forms, inner products |

15A66 | Clifford algebras, spinors |

16H05 | Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) |

16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |

18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |