Alg\`ebre commutative M\'ethodes constructives

TL;DR

This book introduces constructive commutative algebra, emphasizing algorithmic methods for classical theorems, especially in finitely generated projective modules and related theories, revisiting classical algebraic structures with a constructive perspective.

Contribution

It provides an explicit algorithmic approach to classical algebraic theories, making existence proofs constructive and revisiting theories like Galois theory and Dedekind rings with new insights.

Findings

Explicit algorithms for existence theorems

Revisiting classical theories with constructive methods

Simplification of complex algebraic structures

Abstract

This book is an introductory course to basic commutative algebra with a particular emphasis on finitely generated projective modules, which constitutes the algebraic version of the vector bundles in differential geometry. We adopt the constructive point of view, with which all existence theorems have an explicit algorithmic content. In particular, when a theorem affirms the existence of an object -- the solution of a problem -- a construction algorithm of the object can always be extracted from the given proof. We revisit with a new and often simplifying eye several abstract classical theories. In particular, we review theories which did not have any algorithmic content in their general natural framework, such as Galois theory, the Dedekind rings, the finitely generated projective modules or the Krull dimension. Constructive algebra is actually an old discipline, developed among others by Gauss and Kronecker. We are in line with the modern "bible" on the subject, which is the book by Ray Mines, Fred Richman and Wim Ruitenburg, A Course in Constructive Algebra, published in 1988.