**Paul Dubreil**'s father was Léon-René Dubreil, a teacher at the lycée Montesquieu in Le Mans. Born in Le Mans on 31 March 1863 in Le Mans, from a family originally from Parcé-sur-Sarthe, Léon trained to be a teacher in Le Mans from 1879 to 1882. He taught at the lycée Montesquieu there from 1882 until his death on 6 April 1924. On his death the following tribute was paid to him (see [1]):-

Paul had an older sister Marie-Rose Dubreil, born in Le Mans on 5 October 1900. Marie-Rose studied at the girls school in Le Mans and went on to study English at the Faculty of Arts of Caen. After teaching at Nice, then at Lorient, she taught at the lycée Montesquieu in Le Mans from 1929 to 1934 before moving to Paris where she obtained a doctorate on the topic of the novelist Joseph Conrad.He was everywhere loved by his students, with his kindness, poorly concealed by apparent rudeness, joined by bonds of shared affection.

Léon-René Dubreil's second child was Paul Dubreil, the subject of this biography, who was three and a half years younger than his sister. He was taught at home by his parents before entering a primary school in Le Mans in 1907. After completing his primary schooling, he attended the Lycée in Le Mans where his father was the professor of mathematics. He continued to study there until 1921. Of course, World War I took place from 1914 to 1918 while Dubreil was at school in Le Mans. His father was not conscripted in 1914 since he was too old to serve in the military and continued to teach in Le Mans. The war did, however, have a huge impact on the schools in Le Mans. The lycée Montesquieu was, in part, requisitioned to serve as a hospital and 100 wounded soldiers arrived there on 16 August 1914. The schools continued to operate throughout the war years but many of the teachers were mobilised and many were killed in action. These were difficult times for the young Dubreil who was fourteen years old when the war ended.

Graduating from the lycée Montesquieu in 1921, Dubreil went to Paris where he studied at the Lycée St Louis, preparing for the entrance examinations to one of the École Normale Supérieure or the École Polytechnique. He took the examinations in July 1923 and was ranked first for admission to the École Normale Supérieure and second for admission to the École Polytechnique. He chose to study at the École Normale Supérieure, beginning his studies there later that year. At the École Normale Supérieure, in the first semester, he attended lectures by Ernest Vessiot on Galois theory which was presented in a similar way to that in Émile Picard's *Traité d'analyse* . In the second semester of his first year, 1923-24, he attended a course of analysis given by Émile Picard which looked at topics such as plane algebraic curves and abelian integrals. He also used the textbook *Théorie des Fonctions algébriques de deux Variables indépendantes* by Émile Picard and Georges Simart. In his first year of study, he also attended some sessions of Vessiot's seminar on the geometry of numbers according to Minkowski and some sessions of Hadamard's seminars but Dubriel writes in [5] that his:-

Of his first year of study Dubreil wrote, rather modestly, in [5]:-... training was insufficient to draw real profit.

Not only did Dubreil have outstanding teachers at the École Normale Supérieure, but he also had some excellent mathematicians as fellow students including René de Possel and Pierre Honnorat. These two fellow students became very interested in Camille Jordan'sPersonally, I found it hard and obtained a rather mediocre result in the examination. But I remember having experienced great enthusiasm following the Émile Picard's lectures on Riemann surfaces ...

*Traité des substitutions*as well as George-Henri Halphen's paper

*Mémoire sur la classification des corbes gauches algébriques*and Max Noether's paper

*Zur Grundlegung der Theorie der algebraischen Ramcurven*.

In 1924-25 Dubreil attended the Cours Peccot delivered at the Collège de France by Marcel Légaut (1900-1990) on *Etude géométrique des systèmes de points dans un plan. Application à la théorie des courbes gauches algébriques* . Dubreil also read Légaut's 1926 translation into French of *Courbes et fonctions algébraiques d'une variable* by Federigo Enriques. He went to the Sorbonne to complete his License es Sciences. In the 1926 Agrégation de Mathématiques, a national examination to find the best candidates for teaching positions, Dubreil was placed first in the whole of France. Before continuing with his studies, however, he undertook the compulsory military service.

In October 1927, after completing his military service, he was appointed as a lecturer at the École Normale Supérieure and he began working towards his doctorate. He was undertaking research on algebraic geometry but soon encountered difficulties [5]:-

In 1929 he won a prestigious Rockefeller scholarship which enabled him to visit Hamburg in October of that year to study with Emil Artin. He attended Artin's lectures, given twice a week, from 11:15 to 12:00 and 12:15 to 13:00. After the lectures, Artin would return to the Institute followed by his students, including Dubreil, and after chatting until around 14:00, they all went with Artin to a nearby restaurant. There they would discuss mathematics but there were also other topics of conversation such as literature, music and the cinema. Emmy Noether, invited by the Philosophical Society, visited Hamburg in February 1930. She attended Artin's lecture in the morning of the day she was to lecture and came to the lunch which followed. Dubreil got to know her at the lunch, then he attended her lecture in the evening which was given in the main lecture theatre to an audience of philosophers and mathematicians. He wrote to Marie-Louise Jacotin, whom he was to marry in a few months time, and gave her his impressions of Emmy Noether [6]:-I soon had the feeling of not being really good at it: I saw trouble everywhere, I was unable to build acceptable reasoning. ... Such were my tribulations when I had a conversation with my friend André Weil who had just returned from a visit to Göttingen(probably as a Rockefeller Fellow)where he had met Emmy Noether(the daughter of Max Noether)and van der Waerden. He pointed out to me that I was immersed, without knowing it, in the theory of ideals of polynomials and advised me to read the memoir of van der Waerden 'Zur Nullstellen der Polynomideale' and introduced me to the fundamental work of Emmy Noether and of Wolfgang Krull. Reading these works, which were clear and rich in new ideas, gave me enthusiasm. In July1928, my thesis was almost finished ...

Dubreil, however, found discussions with her extremely useful. For the summer semester, beginning at the end of April, Dubriel worked with Noether. However, before that he took a vacation in Groningen so that he could meet Bartel van der Waerden. Then he went to Frankfurt because by that time Noether was in Frankfurt for the summer semester, visiting Carl Siegel, and Dubreil wanted to continue exchanging ideas with her. As soon as he arrived in Frankfurt, Noether told him that she wanted him to talk at the seminar in the next semester. In Frankfurt, Dubreil also met Max Dehn and his students Wilhelm Magnus and Ruth Moufang. He returned to Paris to marry Marie-Louise Jacotin on 28 June 1930. She was also an outstanding mathematician and the two cooperated for the rest of their careers. After their marriage, they went together to Frankfurt arriving there just before the summer semester ended towards the end of July. Dubreil returned to Paris briefly to defend his doctoral thesisI saw a small plump woman with a ruddy complexion and with no sartorial elegance. Obviously very intelligent, she talked a lot, very quickly, in a jerky way. She fell headlong into the trap that this was a conference for philosophers and mathematicians and her lecture was incomprehensible to philosophers.

*Recherches sur la valeur des exposants des composants primaires des ideaux de polynomes*in October 1930. Certainly Dubreil was going to get the most out of his Rockefeller scholarship and he next went to Rome where he discussed problems with the geometers Guido Castelnuovo, Federigo Enriques and Francesco Severi. In Rome, Dubreil and his wife lived in the Hotel Vittoria on the Via Sardegna in the Pincio district. Federigo Enriques and Tullio Levi-Civita were also living there at this time. While in Rome, Dubreil was invited to a dinner given by Levi-Civita in honour of Solomon Lefschetz who, along with his wife, was spending a sabbatical year in Europe.

Dubreil's main interest at that time was algebraic varieties and he believed that he had learnt most from NoetherNoether so, before returning to France, his final visit was again to Göttingen to visit NoetherNoether. Once there, he met Lars Ahlfors who, like Dubreil, was supported with a Rockefeller scholarship. At this time, he also met Hans Fitting, Hans Heilbronn (who was Edmund Landau's assistant), Kurt Mahler and Helmut Ulm (1908-1975). Several visitors also spoke at Emmy Noether's seminar including [6]:-

Herbrand's accident was reported in... Beniamino Segre, André Weil, who was returning from India, and Jacques Herbrand who made two brilliant presentations. He left us in early July to go mountain climbing in France with some comrades, and it was during one of these mountain climbs that the accident happened from which he died.

*Le Temps*on 29 July 1931 and a cutting from the paper sent to Dubreil in Göttingen. He told Emmy Noether what had happened and she was deeply saddened and kept repeating, "That is unthinkable." While in Göttingen, Dubreil joined in the traditional walk after the Seminar in the wooded hills east of the city. Emmy Noether would also take part in the walks which would stop at small rural restaurants providing a variety of attractions.

The exciting travels of his fellowship over, Dubreil took up his first permanent post at the University of Lille. Two years later, in 1933, he moved to Nancy where he was to spend the period of World War II. During this period his wife Marie-Louise had appointments at Rennes and then at Poitiers. In November 1946 Dubreil returned to the Sorbonne and there, in October 1954, he was appointed to the chair of arithmetic and number theory. This chair had been held by Albert Châtelet until he retired in 1954.

After his Ph.D. thesis, Dubreil published a series of papers: *Recherches sur la valeur des exposants des composants primaires des idéaux de polynomes* (1930); *Sur quelques propriétés des systèmes de points dans le plan et des courbes gauches algébriques* (1933); *Sur les intersections totales mixtes dans l'espace à trois dimensions* (1933); *Sur quelques propriétés des variétés algébriques* (1934); and *Quelques propriétés des variétés algébriques se rattachant aux théories de l'algèbre moderne* (1935). He began to work in more general algebraic structures around 1936 when he became interested in generalising the familiar elementary properties of groups into more general settings. He studied the lattice of equivalence relations on sets and from there was led to study semigroups. Writing in 1936 he explained his approach to attacking problems (see for example [12]):-

Axel Thue had published on semigroups as early as 1914 when he had posed word problem for semigroups. However it was not until the late 1930s and early 1940s that the study of semigroups became a major topic. Anatoly Malcev, Alfred Clifford and Dubreil became major figures in the new subject. Dubreil's first major work in this area wasThe questions to which I have tried to make a personal contribution belong both to the field of algebra and that of geometry, and are related to problems which have been studied or posed for some time. If these problems had received only partial solutions, the reason seems to be, regardless of their own difficulties, that they had been considered only from one of the two points of view, algebraic or geometric. Instead, forcing myself to never separate these two points of view, highlighting the geometric meaning of algebraic results and making progress with algebraic methods so that they can meet the needs of geometry, I could not only elucidate points which remained obscure ... but also address problems from a very general and essentially new approach.

*Contribution à la théorie des demi-groupes*(1941); a second part of this paper

*Contribution à la théorie des demi-groupes II*appeared in 1951 and a third part in 1953.

Lallement who was a student of Dubreil's, describes his lectures in [12]:-

In 1946 Dubreil published his bookEach of his lectures was a brilliant piece of exposition, clear, precise, polished, and at the same time inspiring by his ability to relate the topics in hand to past and current research. I remember vividly the keen competition this course[Algebra and Number Theory]generated among students who were all eager to solve the most challenging weekly problems. When I became one of his doctoral students in1961, I realised how deep Paul Dubreil's influence on young algebraists was. He was directing a large number of theses, and the weekly meetings of the Dubreil-Pisot seminar were rallying points of researchers, coming even from distant universities to listen to a very wide variety of guest lectures.

*Algèbre. Tome I. Équivalences, Opérations, Groupes, Anneaux, Corps*. A second edition was published in 1954 and a third edition in 1963. In collaboration with his wife Marie-Louise Dubreil-Jacotin, he published

*Leçons d'algèbre moderne*in 1964. A English edition was published in 1967 under the title

*Lectures on modern algebra*.

Dubreil received many honours for his outstanding contributions, the most notable of which are the Grand Prix des Sciences Mathématiques of the Académie des Sciences (1952)

and being made an Officer of the 'Ordre national de la Légion d'honneur' (1955).... for all his work in algebra and analysis

**Article by:** *J J O'Connor* and *E F Robertson*