Died: 28 June 1984 in Paris, France

**Claude Chevalley** was the only son of Abel and Marguerite Chevalley who were the authors of the Oxford Concise French Dictionary. He studied under Émile Picard at the École Normale Supérieur in Paris, graduating in 1929. After graduating Chevalley continued his studies in Germany, studying under Artin at Hamburg during session 1931-32. He then went to the University of Marburg where he studied under Hasse who had been appointed to fill Hensel's chair there in 1930. He was awarded his doctorate from the University of Paris in 1933 for his thesis *Sur la théorie du corps de classes dans les corps finis et les corps locaux.* He went on to become the youngest of the Bourbaki mathematicians, being a founder member of the group when it formed in December 1934.

In 1938 Chevalley went to the United States to the Institute for Advanced Study at Princeton. When World War II broke out Chevalley reported to the French Embassy in the United States but remained there throughout the war, serving on the Faculty of Princeton University. From July 1949 until June 1957 he served as professor of mathematics at Columbia University, becoming an American citizen during this period. While at Columbia University he tried to return to France, applying for a vacant chair at the Sorbonne. However he ran into severe difficulties in his application for this chair and was unable to return until 1957 when he was appointed to the Université de Paris VII.

Chevalley had a major influence on the development of several areas of mathematics. His papers of 1936 and 1941 where he introduced the concepts of adèle and idèle led to major advances in class field theory and also in algebraic geometry. He did pioneering work in the theory of local rings in 1943, developing ideas due to Krull. Chevalley's theorem was important in applications made in 1954 to quasi-algebraically closed fields and applications made the following year to algebraic groups. Chevalley groups play a central role in the classification of finite simple groups. His name is also attached to Chevalley decompositions and to a Chevalley type of semi-simple algebraic group. He also did fundamental work on spinors which was described by Pierre Cartier and Catherine Chevalley, Chevalley's daughter. They explain that at the time Chevalley carried out the work:-

... spinors were a well-established tool in theoretical physics, and Élie Cartan had already published his account of the theory. But Chevalley's approach to Clifford algebras was quite new in the1950s, at a time where universal algebra was blossoming and developing fast. ... Chevalley's exposition of the algebraic theory of spinors contains a number of interesting innovations. But Chevalley was an algebraist at heart, and gives no hint of the applications to theoretical physics.

Many of Chevalley's texts have become classics and new editions continue to appear as do translations into several different languages. He wrote *Theory of Lie Groups* in three volumes which appeared in 1946, 1951 and 1955. The authors of [7] write:-

Chevalley's most important contribution to mathematics is certainly his work on group theory ...[The Theory of Lie Groups]was the first systematic exposition of the foundations of Lie group theory consistently adopting the global viewpoint, based on the notion of analytic manifold.

G D Mostow, reviewing Volume 2 of the *Theory of Lie Groups* writes:-

The outstanding feature of the exposition is the elegant organization of ideas. The basic definitions are chosen deftly, and each topic is developed with simple directness. Another feature is the meticulous treatment of details which are usually passed over lightly. The book is essentially self-contained and puts the theory on a clear-cut foundation.

Chevalley also published *Theory of Distributions* (1951), *Introduction to the theory of algebraic functions of one variable* (1951), *The algebraic theory of spinors* (1954), *Class field theory* (1954), *The construction and study of certain important algebras* (1955), *Fundamental concepts of algebra* (1956) and *Foundations of algebraic geometry* (1958). Let us quote from reviews of some of these works to give some impression of the material Chevalley covers and the style of his writing.

Zariski reviewing *Introduction to the theory of algebraic functions of one variable* writes:-

In this book the author develops systematically the theory of fields R of algebraic functions of one variable over arbitrary fields K of constants .... The approach is therefore very general, and the treatment incorporates most of the new ideas and methods which have been introduced into the purely algebraic theory of function fields since the appearance of the classical treatise of Hensel-Landsberg. The manner in which the classical material is developed and adapted to arbitrary fields of constants is novel in many respects, and almost every chapter shows distinct traces of the author's original thinking about the subject.

It is worth noting that Weil was a believer in the same Bourbaki style of mathematics as Chevalley, yet when reviewing the same work described it as a:-

... severely dehumanised book.

Kaplansky writes of *Fundamental concepts of algebra:-*

A generation of algebraists grew up for whom "modern algebra" meant Van der Waerden's book, or possibly one of several similar later texts. Time has passed and(happily)mathematics has not stood still. In particular algebraic topology has exhibited an insatiable appetite for algebraic gadgets. In response, modern algebra has changed. What distinguishes the new modern algebra from the old? The latter emphasized groups, rings, and homomorphisms as the basic concepts. Modules, more or less sitting astride groups and rings, were prominent, though perhaps not sufficiently prominent. But at least two things, now clearly of central importance, were completely missing: the tensor product of modules, and the generalization of every object to a graded object. Chevalley's book is timely and it will be widely studied; the meaty exercises will invite a diligent reader to educate himself some more. Teachers may find it "futile to disguise the austerity"(last sentence of the preface).

Chevalley's daughter, Catherine Chevalley, wrote about her father in "Claude Chevalley described by his daughter" (1988):-

For him it was important to see questions as a whole, to see the necessity of a proof, its global implications. As to rigour, all the members of Bourbaki cared about it: the Bourbaki movement was started essentially because rigour was lacking among French mathematicians, by comparison with the Germans, that is the Hilbertians. Rigour consisted in getting rid of an accretion of superfluous details. Conversely, lack of rigour gave my father an impression of a proof where one was walking in mud, where one had to pick up some sort of filth in order to get ahead. Once that filth was taken away, one could get at the mathematical object, a sort of crystallized body whose essence is its structure. When that structure had been constructed, he would say it was an object which interested him, something to look at, to admire, perhaps to turn around, but certainly not to transform. For him, rigour in mathematics consisted in making a new object which could thereafter remain unchanged.

The way my father worked, it seems that this was what counted most, this production of an object which then became inert-dead, really. It was no longer to be altered or transformed. Not that there was any negative connotation to this. But I must add that my father was probably the only member of Bourbaki who thought of mathematics as a way to put objects to death for aesthetic reasons.

Pierre Cartier writes in [2]:-

Chevalley was a member of various avant-garde groups, both in politics and in the arts. As the editor of Chevalley's work, I have decided, at the urging of his daughter, to include a special volume about his work outside mathematics. He had written various pamphlets, and various notes; Catherine Chevalley will have to work hard to collect these things and we will publish them as part of his collected works. ... Mathematics was the most important part of his life, but he did not draw any boundary between his mathematics and the rest of his life. Maybe this was because his father was an ambassador, so he had more contact with various people.

Chevalley was awarded many honours for his work. Among these was the Cole Prize of the American Mathematical Society awarded to him in 1941 for his paper *La théorie du corps de classes* published in the *Annals of Mathematics* in 1940. Chevalley was elected a member of the London Mathematical Society in 1967.

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

**December 2005**

[http://www-history.mcs.st-andrews.ac.uk/Biographies/Chevalley.html]