Misc,
Algèbre commutative Meéthodes constructives
(Nov 8, 2016)
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.
