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A course in constructive algebra. (English) Zbl 0725.03044

Universitext. New York etc.: Springer-Verlag. xi, 344 p. DM 68.00 (1988).
This is the first textbook in Constructive Algebra covering a wide part of the subject matter. The general standpoint of the book is Bishop’s Constructive (or Explicit) Mathematics [for a general ‘constructivist manifesto’ see E. Bishop, Foundations of constructive analysis (1967; Zbl 0183.015)]. The fundamental purpose of Constructive Mathematics is not to build a system alternative to classical mathematics (such as intuitionism, which leads to classically false theorems) nor to refer only to some particular class of ‘constructive’ objects (such as recursive functions) but to extract from classical mathematics its constructive portion. In this sense constructive algebra is nothing but generalized algebra (i.e. standard algebra built in a weaker logical framework). In other words, classically false propositions cannot be proved within this context.
The book covers a wide portion of algebra; after an introductory part (regarding the constructive notion of set and the basic algebraic concepts) the successive chapters are devoted to rings and modules, divisibility in discrete domains, principal ideal domains, field theory, factoring polynomials, commutative Noetherian rings, finite-dimensional algebras, free groups, Abelian groups, valuation theory and Dedekind domains.
The book may be regarded as self-contained, even though some familiarity with the classical subject is presumed at both the student’s and the researcher’s level. Some familiarity with the general concepts, methods and purposes of Constructive Mathematics may be useful as well, in order to make smoother the impact with the first chapters of the book.

MSC:

03F65 Other constructive mathematics
13-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra
03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations

Citations:

Zbl 0183.015
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