# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Lectures on division algebras. (English) Zbl 0934.16013
Regional Conference Series in Mathematics. 94. Providence, RI: American Mathematical Society (AMS). vii, 119 p. \$ 22.00 (1999).
The author’s aim in these lectures is to present the foundations of the modern theory of division algebras and provide a solid background for the massive research done in recent years. He begins with definitions and examples but soon introduces Azumaya algebras and many of the later results are stated in this context. Thus he describes the ‘form of matrices’ obtained by making a faithfully flat extension of the ground ring. The Brauer group is defined for any commutative ring $R$, though for its more recondite properties $R$ is taken to be a field. Next crossed products are introduced and are used to describe the equivalence between Brauer groups and certain Galois cohomology groups. This leads to the corestriction and the Merkurjev-Suslin theorem which is quoted without proof. After a brief discussion of orders, lattice methods and ramification the author comes to lifting properties and specializations, leading to the notion of a generic division algebra, which is discussed in some detail and is related to the Brauer-Severi variety. The style of the whole is very informal, not to say casual. This sometimes makes for better understanding but occasionally leads to confusion (much like a real lecture). Only a basic background is expected from the reader, e.g. some proofs depending on étale cohomology are omitted, with just a reference. A serious lack is the omission of an index and a list of notations, but this the reader can make for himself. In all this is a useful survey of the present state of the subject, going well beyond most other texts in this field.

##### MSC:
 16K40 Infinite dimensional and general division rings 16K50 Brauer groups 16H05 Separable associative algebras 12G05 Galois cohomology 16-02 Research monographs (associative rings and algebras) 12-02 Research monographs (field theory)