Claude Quitté

Constructive Krull Dimension. I: Integral Extensions.

We give a constructive approach to the well known classical theorem saying that an integral extension doesn’t change the Krull dimension.

Univariate polynomial factorization over finite fields

Abstract This paper is a tutorial introduction to univariate polynomial factorization over finite fields. We recall the classical methods that induced most factorization algorithms (Berlekamps and the Cantor-Zassenhaus ones) and some refinements which can be applied to these methods. Explicit algorithms are presented in a form suitable for almost immediate implementation. We give a detailed description of an efficient implementation of the Cantor-Zassenhaus algorithm used in the release 2 of the Axiom computer algebra system.

Curves and coherent Prüfer rings

We show, in constructive mathematics, that if k is a discrete field and f an arbitrary polynomial in kx,y] then the localisation Rfy is always a semihereditary ring, where R denotes the ring kx,y] quotiented by f. An important corollary is that R is semiherditary whenever 1=?f,fx,fy?. This can be seen as the constructive content of the theorem saying that if moreover R is a domain, then it is Dedekind.

Finitely Presented Modules

Over a ring the finitely presented modules play a similar role as that of the finite dimensional vector spaces over a field: the theory of finitely presented modules is a slightly more abstract, and at times more profitable, way to approach the subject of systems of linear equations. We provide the basics of the theory of finitely presented modules. We treat the examples of finitely presented modules over PIDs, over Bezout rings and over zero-dimensional rings. We introduce important invariants that are Fitting ideals, and last section introduces the resultant ideal as a direct application of the Fitting ideals.

Comparison of Picard groups in dimension 1

We compare two Picard groups in dimension 1. Our proofs are constructive and the results generalize a theorem of J. Sands [11]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Suslin’s Stability Theorem, the Field Case

We give an entirely constructive treatment of Suslin’s stability theorem for the case of discrete fields. We use the concrete local-global techniques explained in chapter XV in order to obtain constructive proofs by dynamic and constructive decryption of the classical proofs.

Finitely Generated Projective Modules, 1

We continue the study of finitely generated projective modules started in Chap. V. In Sect. 1 we readdress the question regarding the characterization of finitely generated projective modules as locally free modules. Section 2 is dedicated to an elementary theory of the rank which does not require prime ideals. In Sect. 3 we give some simple applications of the local structure theorem. Section 4 is an introduction to Grassmannians. In Sect. 5 we introduce the fundamental and difficult problem of classifying finitely generated projective modules over a fixed ring. Section 6 presents a nontrivial example for which this classification can be obtained.

Extended Projective Modules

We constructively establish a few important results regarding the situations where the finitely generated projective modules over a polynomial ring are extended from the base ring. We especially treat Traverso-Swan’s theorem (rank one projective modules over R[X] are extended if and only if R is seminormal), the patching a la Vaserstein-Quillen, Horrocks’ theorems, Quillen-Suslin’s theorem, Bass theorem (finitely generated projective modules over V[X], V a valuation domain are extended) and the Lequain-Simis theorem (generalization of Bass’ theorem for several variables and for arithmetic rings).

Strictly Finite Algebras and Galois Algebras

This chapter begins with the study of finite algebras over a field. A pertinent generalization of this notion to commutative rings is given by the algebras which are finitely generated projective modules over the base ring, we call them strictly finite algebras. Section 3 is a brief introduction to finitely presented algebras, by insisting on the case of algebras which are integral over the base ring. In Sects. 5 and 6, we introduce the neighboring notions of strictly etale algebra and of separable algebra. In Sect. 7 we give a constructive presentation of the Galois algebra theory for commutative rings.

Distributive Lattices Lattice-Groups

This chapter begins with an introductory section which fixes the formal algebraic framework of distributive lattices and of Boolean algebras.