Böcker av Francis Borceux

This textbook arises from a master's course taught by the author at the University of Coimbra. It takes the reader from the very classical Galois theorem for fields to its generalization to the case of rings. Given a finite-dimensional Galois extension of fields, the classical bijection between the intermediate field extensions and the subgroups of the corresponding Galois group was extended by Grothendieck as an equivalence between finite-dimensional split algebras and finite sets on which the Galois group acts. Adding further profinite topologies on the Galois group and the sets on which it acts, these two theorems become valid in arbitrary dimension. Taking advantage of the power of category theory, the second part of the book generalizes this most general Galois theorem for fields to the case of commutative rings. This book should be of interest to field theorists and ring theorists wanting to discover new techniques which make it possible to liberate Galois theory from its traditional restricted context of field theory. It should also be of great interest to category theorists who want to apply their everyday techniques to produce deep results in other domains of mathematics.

This third volume in a trilogy of texts on geometry guides students through the development of differential geometry. It links classical surface theory with modern Riemannian geometry and prepares readers for advanced topics such as algebraic topology.

This book offers a unified treatment of the various algebraic approaches of geometric spaces. It details the algebraic ingredients necessary to develop all the major aspects of the theory of algebraic curves.

This volume's historical focus, with fully worked solutions to all the famous problems in classical geometry, demonstrates the profound influence of axiomatic geometry, over more than three millennia, on the evolution of mathematics as an academic discipline.