## Reviews of books by Paul Dubreil

**1. Algèbre. Tome I. Équivalences, Opérations, Groupes, Anneaux, Corps (1946), by Paul Dubreil.**

**1.1. Review by: Garrett Birkhoff.**

*Mathematical Reviews*MR0017734

**(8,192k)**.

This lucid and simple introduction to abstract algebra approaches the subject from an extremely general point of view. Thus it begins with a general discussion of sets and relations. The concepts of equivalence relations and of "associable'' or permutable equivalence relations are also treated in general. Various postulate systems for groups and their generalizations (groupoids, demigroups, etc.) are compared at length. Right-and left-substitution and cancellation laws are discussed. Ordered fields and domains are also discussed very carefully, with especial reference to completeness properties. The book concludes with an exposition of unique factorization, Euclidean domains, polynomial rings, perfect fields, symmetric polynomials and related topics.

**1.2. Review by: Alonzo Church.**

*The Journal of Symbolic Logic*

**12**(3) (1947), 94.

This is an introduction to abstract algebra. Within the field of the Journal of Symbolic Logic are brief informal expositions (in the first chapter and the first two appendices) of class algebra, relations, equivalence relations, definitions of finiteness, arithmetic as based on Peano's postulates, the axiom of choice.

**2. Algèbre. Tome I. Equivalences, opérations, groupes, annaux, corps (2nd ed.) (1954), by Paul Dubreil.**

**2.1. Review by: M P Drazin.**

*The Mathematical Gazette*

**41**(335) (1957), 61-62.

Volume I is concerned with fundamentals: the emphasis is on sets, relations, correspondences, operations and "regular equivalences", while the remaining topics listed in the chapter headings comprise groups, semigroups, fields, rings and algebraic equations. The first chapters include a very full treatment of the general theory of relations, special attention being given to partial and total orderings (and to their link with lattice theory). No previous knowledge is assumed, but the author expects a certain mathematical maturity in his reader, who is nowhere patronised, except perhaps by a completeness which leaves little to his imagination (even the slightest of theorems being accorded the dignity of a full-dress proof). At the beginning this might be welcomed by a reader not yet accustomed to abstract thought, but it is surely out of place in the later pages. And such a reader without previous algebraic experience is in any case hardly more than a polite fiction: though the book is, technically, self-contained, it is essentially a polished and unified (if somewhat discursive) treatment for those already familiar, or at least acquainted, with what it unifies. The author's extremely general and abstract approach marks him out as an Algebraists' Algebraist, and, in the reviewer's opinion, the real beginner should look elsewhere for his first, and inevitably partial, glimpses of the structures whose broader perspectives are so well exhibited here. ... If one regrets being slowed down by over-detailed trivialities, one has nevertheless to be thankful for the crystal-clear discussions of deeper questions; similarly, a sustained freshness of approach more than compensates for a few slightly mystifying unconventionalities. One can often guess with tolerable accuracy what an author of lesser calibre will be saying for several pages ahead. But this book follows none of the well-worn grooves. The author has interesting points to make even on the most standard topics, and the book is full of illuminating comments (usually as footnotes) on matters which sometimes cause confusion.

**2.2. Review by: R E Johnson.**

*Mathematical Reviews*MR0064746

**(16,328g)**.

The various topics treated in the first edition (1946) have been brought up to date in this new edition. Also, some new topics have been added. The first chapter now has a complete discussion of relations, as well as a new section on ordered sets and lattices. The fifth chapter on ordered fields has been enlarged to include more material on the imbedding problem for demi-groups.

**3. Les relations d'équivalence et leurs principales applications (1954), by Paul Dubreil.**

**3.1. Review by: D C Murdoch.**

*Mathematical Reviews*MR0067083

**(16,667g)**.

This article contains a brief survey of the theory of equivalence relations, first in an arbitrary set and then in algebraic systems. Here the concept of regularity of an equivalence relation relative to an algebraic operation leads to the general homomorphism theorem.

**4. Problèmes de Mesure (1962), by H Cartan, J Dixmier, P Dubreil, A Lichnerowicz and A Revuz.**

**4.1. Review by: R P Gillespie.**

*The Mathematical Gazette*

**48**(364) (1964), 243-244.

The main part of the book and the part properly described by the title consists of the lectures on 'Volumes of Polyhedra' by Henri Cartan, on 'Measure of Angles' by J Dixmier and on the 'Theory of Integration' by A Revuz. To this is added a historical lecture on the 'Development of Algebra' by Paul Dubreil and one on 'Tensor Calculus' by Andre Lichnerowicz. ... The fourth lecture is an outstanding historical account of the part played by the problem of the algebraic solution of equations in the development of the ideas of modern abstract algebra. The work in turn of Lagrange, Abel, Galois, Jordan and Steinitz is explained in a most lucid manner. Along with the discussion of the mathematics, the author describes something of the lives of the mathematicians. His account of the disappointments and sufferings of Abel and Galois is most moving.

**4.2. Review by: Paul R Halmos.**

*Mathematical Reviews*MR0159917

**(28 #3133)**.

This is a set of five expository articles, two of which have nothing to do with measure. ... Dubreil gives a historical account of some aspects of algebra from Lagrange to Steinitz, via Abel and Galois.

**5. Algèbre. Tome I. Equivalences, opérations, groupes, annaux, corps (3rd ed.) (1963), by Paul Dubreil.**

**5.1. Review by: Editors.**

*Mathematical Reviews*MR0158923

**(28 #2145)**.

A revision of a text on the algebraic concepts mentioned in the title; an earlier edition (1946) was reviewed.... Table of contents: Chapter I, Ensembles, Relations, Correspondances; Chapter II, Opérations; Chapter III, Groupes; Chapter IV, Équivalences régulières; Chapter V, Demigroupes, Corps; Chapter VI, Anneaux; Chapter VII, Équations algébriques; Note I, Ensembles finis; Note II, Nombres naturels; Note III, Axiome du choix et les théorèmes de Zermelo et de Zorn; Note IV, Espaces; Note V, Correspondances biunivoques.

**6. Leçons d'algèbre moderne (1964), by Paul Dubreil and Marie-Louise Dubreil-Jacotin.**

**6.1. Review by: R E Jenner.**

*Mathematical Reviews*MR0166248

**(29 #3525)**.

This work was written as a textbook for an already existing course, the content of which was apparently preordained. Possibly for this reason the book is somewhat conservative; indeed, it really could have been written twenty years ago. Nevertheless, it is good. As one would expect from the authors, unusual attention is given to the study of general multiplicatively closed systems. This helps give the book its unique character and effectively illustrates the use of the basic axioms. The style is severe, light on motivation, and with some tendency to prove things in settings considerably more general than their main realm of applicability would justify to many people.

**7. Lectures on modern algebra (1967), by Paul Dubreil and Marie-Louise Dubreil-Jacotin.**

**7.1. Review by: John D P Meldrum.**

*The Mathematical Gazette*

**53**(384) (1969), 207-208.

This book is a translation from the French of a textbook written to cover the complete algebra honours course for undergraduates. ... The approach is different from that most commonly used in similar textbooks, and seems quite a good one. The ground covered is quite substantial and this would be quite a useful textbook for someone doing an Honours course in Pure Mathematics, or who wished to have a grounding in Modern Algebra. It tends to go further than the standard introductory book on modern algebra without becoming too difficult. There are two main criticisms, though. The first is that, though this book is meant for students, it has no exercises. This detracts from its value, as doing some exercises is the best way of consolidating new knowledge. The other is the large number of misprints, many of which are harmless, but some of which could cause a lot of confusion in the mind of an inexperienced mathematician.

**7.2. Review by: Editors.**

*Mathematical Reviews*MR0220551

**(36 #3609)**.

English translation of the second French edition.

JOC/EFR October 2016

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