Skew fields. Theory of general division rings.

*(English)*Zbl 0840.16001
Encyclopedia of Mathematics and Its Applications. 57. Cambridge: Cambridge Univ. Press. xv, 500 p. (1995).

The methods used in the study of skew fields (or division rings) sharply diverge according to whether the skew field is assumed to be finite-dimensional over its center or not. The present monograph makes the sum of knowledge on the methods which apply in the general case. Most of the problems fall under two main headings: fields of fractions and extensions of skew fields. Decisive advances on both kinds of problems were made in the 60’s and 70’s, and the author’s essential contributions were discussed in his lecture notes “Skew field constructions” [Lond. Math. Soc. Lect. Note Ser. 27 (1977; Zbl 0355.16009)]. Recent developments lead to simplifications of the original proofs, allowing the virtually self-contained comprehensive treatment presented here in a highly polished form.

The first three chapters give an exposition of various classical constructions such as Ore rings of fractions, skew polynomial rings and Malcev-Neumann formal power series rings, and of the Galois theory of skew fields based on the Jacobson-Bourbaki correspondence, with applications to left or right polynomial equations over skew fields.

Fields of fractions are studied at length in chapter 4, where a full proof of the existence of a universal field of fractions for a semifir is given. In chapter 5, it is proved that every family of skew fields having a common subfield can be embedded in a universal fashion in a skew field. This fundamental amalgamation property is derived from Bergman’s coproduct theorems, which show (in particular) that a coproduct of skew fields is a fir. Examples of skew field extensions of different left and right degrees are also given in this chapter.

Polynomial and rational identities come to the forefront in the next three chapters. Amitsur’s theorem which asserts that a skew field of infinite degree over its center, itself infinite, has no nontrivial rational identities is proved twice: in chapter 6 it appears as a consequence of the discussion of free skew fields, and in chapter 7 it is the centerpiece of a general treatment of rational identities which also investigates generic division algebras. Connections with logic are explored in chapter 6, where existentially closed skew fields are introduced and the word problem is discussed (and solved for free skew fields). The theme of noncommutative analogues of algebraically closed fields is taken up again in chapter 8, where the technical problems which need to be overcome to develop a noncommutative algebraic geometry are discussed. In the final chapter 9, valuations and orderings on skew fields are discussed from a general point of view.

Each section ends with a substantial collection of exercises, some of which point to additional developments. The notes and comments which conclude each chapter provide precious insights into the historical background of the theory.

The first three chapters give an exposition of various classical constructions such as Ore rings of fractions, skew polynomial rings and Malcev-Neumann formal power series rings, and of the Galois theory of skew fields based on the Jacobson-Bourbaki correspondence, with applications to left or right polynomial equations over skew fields.

Fields of fractions are studied at length in chapter 4, where a full proof of the existence of a universal field of fractions for a semifir is given. In chapter 5, it is proved that every family of skew fields having a common subfield can be embedded in a universal fashion in a skew field. This fundamental amalgamation property is derived from Bergman’s coproduct theorems, which show (in particular) that a coproduct of skew fields is a fir. Examples of skew field extensions of different left and right degrees are also given in this chapter.

Polynomial and rational identities come to the forefront in the next three chapters. Amitsur’s theorem which asserts that a skew field of infinite degree over its center, itself infinite, has no nontrivial rational identities is proved twice: in chapter 6 it appears as a consequence of the discussion of free skew fields, and in chapter 7 it is the centerpiece of a general treatment of rational identities which also investigates generic division algebras. Connections with logic are explored in chapter 6, where existentially closed skew fields are introduced and the word problem is discussed (and solved for free skew fields). The theme of noncommutative analogues of algebraically closed fields is taken up again in chapter 8, where the technical problems which need to be overcome to develop a noncommutative algebraic geometry are discussed. In the final chapter 9, valuations and orderings on skew fields are discussed from a general point of view.

Each section ends with a substantial collection of exercises, some of which point to additional developments. The notes and comments which conclude each chapter provide precious insights into the historical background of the theory.

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

##### MSC:

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

16K40 | Infinite-dimensional and general division rings |

16S10 | Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) |

16E60 | Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc. |

16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |

16S35 | Twisted and skew group rings, crossed products |

16U20 | Ore rings, multiplicative sets, Ore localization |

12E15 | Skew fields, division rings |

12E05 | Polynomials in general fields (irreducibility, etc.) |