The main goal of this book is to find the constructive content hidden in abstract proofs of
concrete theorems in Commutative Algebra especially in well-known theorems concerning
projective modules over polynomial rings (mainly the Quillen-Suslin theorem) and syzygies of
multivariate polynomials with coefficients in a valuation ring. Simple and constructive proofs
of some results in the theory of projective modules over polynomial rings are also given and
light is cast upon recent progress on the Hermite ring and Gröbner ring conjectures. New
conjectures on unimodular completion arising from our constructive approach to the unimodular
completion problem are presented. Constructive algebra can be understood as a first
preprocessing step for computer algebra that leads to the discovery of general algorithms even
if they are sometimes not efficient. From a logical point of view the dynamical evaluation
gives a constructive substitute for two highly nonconstructive tools of abstract algebra: the
Law of Excluded Middle and Zorn's Lemma. For instance these tools are required in order to
construct the complete prime factorization of an ideal in a Dedekind ring whereas the
dynamical method reveals the computational content of this construction. These lecture notes
follow this dynamical philosophy.
