Constructive algebra (Q2878070)

The article under review provides a general introduction to some of the principles, methods and results of constructive commutative algebra, that is, to purely algorithmic approaches to various fundamental existence theorems in the classical theory of commutative rings, fields, ideals, and modules. This particular methodological viewpoint of constructive algebra, which was already developed in the 19th century by C. F. Gauss, L. Kronecker, and other great mathematicians at that time, is masterly elaborated and presented in the recent comprehensive textbook [the second author and \textit{C. Quitté}, Commutative algebra. Constructive methods. Mathématiques en Devenir. Paris: Calvage et Mounet. xxxi, 991 p. (2011; Zbl 1242.13002)]. Actually, this voluminous text may be regarded as a modern continuation of the classic primer [\textit{R. Mines} et al., A course in constructive algebra. Universitext. New York etc.: Springer-Verlag. xi, 344 p. (1988; Zbl 0725.03044)], and is the basic reference for the current brief survey of some fundamental aspects of the subject.NEWLINENEWLINE As for the precise contents of the present article, the authors describe the underlying logic and philosophy of constructive proofs in classical abstract algebra by means of several instructive examples, and they illustrate how constructive methods can help simplify various classical, originally transfinite proofs considerably. Among the examples discussed in the course of the explanations are the computational aspects of the Quillen-Suslin theorem on projective modules over polynomial rings, the so-called ``Positivstellensatz'' of Krivine-Stengle, the characterization of semi-normal rings by Traverso-Swan, and a few further results in this context. The article ends with the authors' conclusion stating that the constructive approach to classical algebra does represent an effective alternative to the usual systematic treatment via the abstract Hilbert program in mathematics, at least so in many concrete cases.NEWLINENEWLINE In fact, this expository article provides a very useful introduction to the basic ideas of constructive algebra, and to the comprehensive textbook by the second author and Quitté [loc. cit.] as well.