Finite-dimensional division algebras over fields.

*(English)*Zbl 0874.16002
Berlin: Springer. x, 278 p. (1996).

The title of this monograph adequately describes its main topic, which includes finite-dimensional central simple algebras and the Brauer group of fields. Fundamental results, such as Wedderburn’s structure theorems and the Skolem-Noether theorem, are assumed; they can be found in the author’s Basic Algebra II [2nd ed. Freeman, New York (1989; Zbl 0694.16001)]. The exposition focuses on a selection of more advanced topics, with particular attention to delicate structure theorems on division algebras of low dimension.

The first two chapters discuss the more classical aspects. Chapter I deals with the construction of division algebras by means of skew polynomial rings and (generalized) cyclic algebras. It includes a useful discussion of the arithmetic of non commutative principal ideal domains. Chapter II introduces the Brauer group of a field and relates it to Galois cohomology through the crossed product construction. A remarkable feature of the exposition is the welcome return to some original methods which had been by-passed by modern treatments. The reader will thus (re)discover with interest Brauer factor sets and Brauer’s result on the splitting of division algebras of degree 5.

The last three chapters contain more recent material. Chapter III yields the first thorough discussion in book form of generic splitting fields of central simple algebras, which have been increasingly used in the last decades. Geometric aspects (Brauer-Severi varieties) are considered as well as purely algebraic aspects, and several constructions of generic splitting fields are discussed. Very little background in algebraic geometry is required, as the author gives a detailed and very explicit account of Grassmannians. Chapter IV deals with \(p\)-algebras, i.e. central simple algebras of degree \(p^n\) over a field of characteristic \(p\) (prime). Albert’s classical theorems are complemented by results of Witt and Saltman on the Galois groups of maximal subfields and by the Amitsur-Saltman construction of non-cyclic \(p\)-algebras.

Chapter V is the largest, with about one third of the total number of pages. It concentrates on the central simple algebras which have an involution, i.e. an anti-ring-automorphism of order 2. If the involution is linear, these algebras are exactly those which represent the 2-torsion elements in the Brauer group, according to a theorem of Albert. The proof given here follows unpublished work of Tamagawa. If the involution is semi-linear, they represent the kernel of the corestriction map to the fixed subfield of the center, by a theorem of Riehm. However, additional structures can be defined by taking a (fixed) involution into account. The author particularly emphasizes the Jordan algebra structure on the space of symmetric elements. Reduced norms and universal envelopes of these Jordan algebras are discussed. In the case of involutorial central simple algebras of degree 4, reduced norms of Jordan subalgebras of symmetric elements are related to the Albert quadratic forms of the algebras.

The theory of finite-dimensional division algebras has witnessed a very substantial development in the last decade; its ramifications now extend to \(K\)-theory, algebraic geometry and number theory. It is therefore not surprising that a book of this scope does not completely cover the various aspects of the theory. It provides however an invaluable new reference at the graduate level, focusing on the purely algebraic and most classical sides. Unfortunately, no index is provided.

The first two chapters discuss the more classical aspects. Chapter I deals with the construction of division algebras by means of skew polynomial rings and (generalized) cyclic algebras. It includes a useful discussion of the arithmetic of non commutative principal ideal domains. Chapter II introduces the Brauer group of a field and relates it to Galois cohomology through the crossed product construction. A remarkable feature of the exposition is the welcome return to some original methods which had been by-passed by modern treatments. The reader will thus (re)discover with interest Brauer factor sets and Brauer’s result on the splitting of division algebras of degree 5.

The last three chapters contain more recent material. Chapter III yields the first thorough discussion in book form of generic splitting fields of central simple algebras, which have been increasingly used in the last decades. Geometric aspects (Brauer-Severi varieties) are considered as well as purely algebraic aspects, and several constructions of generic splitting fields are discussed. Very little background in algebraic geometry is required, as the author gives a detailed and very explicit account of Grassmannians. Chapter IV deals with \(p\)-algebras, i.e. central simple algebras of degree \(p^n\) over a field of characteristic \(p\) (prime). Albert’s classical theorems are complemented by results of Witt and Saltman on the Galois groups of maximal subfields and by the Amitsur-Saltman construction of non-cyclic \(p\)-algebras.

Chapter V is the largest, with about one third of the total number of pages. It concentrates on the central simple algebras which have an involution, i.e. an anti-ring-automorphism of order 2. If the involution is linear, these algebras are exactly those which represent the 2-torsion elements in the Brauer group, according to a theorem of Albert. The proof given here follows unpublished work of Tamagawa. If the involution is semi-linear, they represent the kernel of the corestriction map to the fixed subfield of the center, by a theorem of Riehm. However, additional structures can be defined by taking a (fixed) involution into account. The author particularly emphasizes the Jordan algebra structure on the space of symmetric elements. Reduced norms and universal envelopes of these Jordan algebras are discussed. In the case of involutorial central simple algebras of degree 4, reduced norms of Jordan subalgebras of symmetric elements are related to the Albert quadratic forms of the algebras.

The theory of finite-dimensional division algebras has witnessed a very substantial development in the last decade; its ramifications now extend to \(K\)-theory, algebraic geometry and number theory. It is therefore not surprising that a book of this scope does not completely cover the various aspects of the theory. It provides however an invaluable new reference at the graduate level, focusing on the purely algebraic and most classical sides. Unfortunately, no index is provided.

Reviewer: J.-P.Tignol (Louvain-La-Neuve)

##### MSC:

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

16K20 | Finite-dimensional division rings |

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

12E15 | Skew fields, division rings |

17C50 | Jordan structures associated with other structures |

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