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Topological fields. (English) Zbl 0683.12014
North-Holland Mathematics Studies, 157; Notas de Matématica, 126. New York etc.: North-Holland. xiii, 563 p. $ 110.50; Dfl. 210.00 (1989).
Table of contents:
I. Topological groups (pp. 1-75)
II. Topological rings and modules (pp. 77-134)
III. Absolute values (pp. 137-203)
IV. Topological vector spaces and their applications (pp. 207-271)
V. Valuations (pp. 275-363)
VI. Locally bounded topologies (pp. 381-466)
VII. Historical notes (pp. 469-502)
As it is seen, one half of the book deals with topological algebras different from topological fields. Thus the title of the book is misleading: it should be rather something like “Treatise on topological algebra”, “Topological algebra” than “Topological fields”.
Chapter I deals with fundamental properties of topological groups, completions, Kakutani’s metrization theorem (6.4), summability. However Pontryagin’s duality and its applications to the topological algebra are not mentioned.
Chapter II presents standard definitions and properties of topological rings and modules, their completions, continuity of inversion in topological rings, locally bounded rings, criteria of normability (Milman- Lipkina- Arnautov- Warner theorem (17.10)), criterion of normability by a power-multiplicative norm - Aurora’s theorem (17.10).
In Chapter III fundamental properties of absolute values and valuations on rings and criteria for a division ring topology to be a valued topology (Shafarevich-Kaplansky theorem) are presented. §§ 21-22 contain p-adic numbers and elementary nonarchimedean analysis.
In Chapter IV topological vector spaces and their applications are given. Out of their elementary properties, in § 25 principles of functional analysis of vector spaces over a valued division ring are presented. In particular, this chapter contains a discussion of extensions of absolute values, Frobenius theorem on division algebras over \({\mathbb{R}}\), Gelfand- Mazur theorem and van Dantzig- Jacobson- Pontryagin- Kowalsky classification of locally compact division rings.
The long (105 pages) Chapter V contains a detailed study of valuations on fields and it could form alone a book.
Chapter VI presents theory of locally bounded topologies on global fields and their characterization given in full generality by Hans Weber. The most essential construction here is that of the local direct sum of rings relative to a family of subrings, i.e. an adelic-like construction. The purely algebraic § 34 presents Dedekind domains. Different forms of Correl- Jebli- Heine- Warner- Cohen results about linear topologies on the quotient field of a Dedekind domain are given.
In Chapter VII the author presents his point of view on the history of topological algebra. His report on the history of the subject is too short to be treated as the history of topological algebra, but is interesting and useful for everybody.
Bibliography contains references mainly about topological fields. Selected papers on topology and topological groups are added.
The book is self-contained and is clearly written. It contains practically no examples. This makes the book not easy to read for non- specialists or beginners. Many interesting results (I do not want to say - the most) are contained in the exercises. There are 389 exercises and many of them are difficult - many original results are included in them. The author wanted to say everything on the presented topics in his book. Consequently, for a non-advanced reader it can be difficult to distinguish between more or less essential notions and results. The author’s style is Bourbaki-like.
Several problems essential for the topological algebra are not discussed, e.g. extension problem for ring topologies on fields, cardinality results and non-standard methods in topological fields are omitted. Sometimes the most essential results in the exercises are indicated by suitable references, but the author does not do this consequently.
For example: on page 40, exercise 5.3 one should add Aurora [1969 c]; similarly on p. 84, exercise 11.7 (a) Kallman [1983 a]; on p. 111, 14.10 theorem: Banach [1948]; on p. 322, exercise 30.20: Kiltinen [1967] (theorem 6.1); exercise 30.21: Kiltinen [1967] (theorem 5.2) and Taǐmanov [1978], etc.
The above criticism does not diminish the value of the book under review. The author’s “Topological fields” can be recommended not only for specialists but for everybody who wants to see a role of topological algebra in contemporary mathematics.
Reviewer: W.Wiȩsław

MSC:
12Jxx Topological fields
13Jxx Topological rings and modules
22-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups
12-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory
12-02 Research exposition (monographs, survey articles) pertaining to field theory
16W80 Topological and ordered rings and modules
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