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Chevalley, Claude

CHEVALLEY, CLAUDE

(b. Johannesburg, Transvaal, South Africa, 11 February 1909;

d. Paris, France, 28 June 1984), algebra, class field theory, group theory.

The most resolute modernizer among the founders of Bourbaki, and the most given to austere axiomatic abstraction, Chevalley was influential in setting the broad agenda of Bourbaki’s project and for major advances in number theory and the theory of Lie groups. He was the chief representative of mathematical logic in France in the 1930s not for any work of his own but for promoting the ideas of his friend Jacques Herbrand, who died in a mountain-climbing accident in 1931 at age twenty-three. Chevalley had philosophical, cultural, and political interests, in which “his mathematician friends had the impression that he proceeded as in mathematics, by the axiomatic method: having posed some axioms he deduced consequences from them by inflexible logic and unconcerned with obstacles along the route which would have led anyone else to go back and change the axioms” (Dieudonné, 1999, p. 113).

Early Life and Education . Chevalley was born into socially rising French Protestant circles little more than a hundred years after Protestants gained full citizenship. His paternal grandfather was a Swiss-born clockmaker naturalized as French. His uncles on that side were a legal councilor to the king of Egypt and a head doctor at l’Hôpital des Enfants-Malades in Paris (France’s first children’s hospital). His father, Daniel Abel Chevalley, passed the agrégation at the École Normale Supérieure de Saint-Cloud to become professor of English at the Lycées Voltaire and Louis-le-Grand in Paris. He became a diplomat in South Africa, Norway, and the Caucasus and Crimea, and returned to academic research after retirement. Chevalley’s maternal grandfather was a village pastor in the Ardèche who became a professor of the Faculty of Protestant Theology of Strasbourg and eventually a founder and the dean of the Faculty of Protestant Theology of Paris. His mother, born Anne Marguerite Sabatier, was also an Anglicist and coauthor with her husband of the first edition of the Concise Oxford French Dictionary. His parents married in 1899 and besides their son had a daughter Lise, who lived to have children but died in 1933. Daniel also died in 1933, while Marguerite lived to 1969. They were active in the Association France-Grande-Bretagne—a group founded during World War I and prominent enough that when the Germans took Paris in 1940 there was barely time to destroy the records before the Gestapo came for them.

Chevalley traveled with his parents until he began school at Chançay and then in Paris at the Lycée Louis-le-Grand, where he gained his love of mathematics and

began to study the standard analysis textbook, Édouard Goursat’s 1902 Cours d’analyse. In 1926 he entered the École Normale Supérieur (ENS) in Paris at the early age of seventeen. He formed an important friendship with Herbrand, who had also been admitted at seventeen, one year before him. The two were drawn to number theory, and to German mathematics, in 1927 when they met André Weil. They took courses from the same excellent but dated mathematicians as Weil. Jacques-Salomon Hadamard’s seminar gave them glimpses of recent mathematics, and they taught themselves more from original sources than they learned in courses. Chevalley studied especially with Émile Picard, graduated in 1929, and published a note on number theory in the Comptes Rendus de l’Academie des Sciences that year. In 1929–1930 Herbrand did his military service and produced the work that Chevalley later said formed the basis of the new methods in class field theory. Herbrand spent 1930–1931 in Germany studying logic, and died on holiday in the Swiss Alps on the way back. Chevalley spent 1931–1932 studying number theory especially with Emil Artin at Hamburg and Helmut Hasse at Marburg and came back to earn his doctorate from the University of Paris in 1933 with a thesis on the work he did in Germany.

Class Field Theory . Class field theory had been at the top of the agenda in number theory since Teiji Takagi around 1920 proved a series of decades-old conjectures of Leopold Kronecker and David Hilbert. The proofs were extremely complicated, and the results were at once productive of concrete arithmetic theorems and promising of further theoretical advances. The subject was prestigious and daunting. Chevalley’s work on it showed a typical difference from Weil, who used modern number theory to expand themes from classical analysis while Chevalley worked to remove analysis in favor of algebra and point-set topology. Weil’s impulse was more classically geometrical and Chevalley’s more algebraic.

Carl Friedrich Gauss already used what are now called the Gaussian numbers Q[i] in arithmetic. These are expressions p+qi where p and q are any ordinary rational numbers and i is the imaginary square root of -1. In particular the Gaussian integers Z[i] are those a+bi where a and b are any ordinary integers. An ordinary prime number may not be prime in the Gaussian integers, as for example the ordinary prime 5 factors as 5 = (2+i)(2-i).

Yet the Gaussian integers do have unique prime factorization analogous to the ordinary integers. This is the prime factorization of 5 as a Gaussian integer. Using other algebraic irrationals in place of i gives other algebraic number fields K in place of the Gaussian numbers Q[i]. And each algebraic number field K contains a ring A of algebraic integers analogous to the Gaussian integers Z[i] although generally not so easy to describe. If rings of algebraic integers always had unique prime factorization, then there would be wonderful consequences, such as a one-page proof of Fermat’s last theorem. They do not. Class field theory began as an astonishing way to measure and work with failures of prime factorization.

Each algebraic number field K extends to a certain larger field L(K) called the Hilbert class field, so that the Galois group of L(K) over K measures the failure of unique prime factorization in the ring A. Furthermore, roughly speaking, all the algebraic integers in A have unique prime factorization in L(K). If A itself has unique prime factorization then K=L(K) and the Galois group is the trivial {1}. A nontrivial Galois group for L(K) shows failure of prime factorization in A, and a larger Galois group shows greater failure. Explicit descriptions of Hilbert class fields were known—some using classical complex analysis. Other arithmetic properties of the ring A are expressed by the Galois groups of other extensions of K, which are also called class fields of various kinds.

The German number theorists would study any given algebraic number field K and its ring of algebraic integers A in connection with other related fields called local fields, so-called because these fields often concentrate attention on a single prime factor. One of Chevalley’s typical contributions was to stress the sense in which every one of them concentrates on a single prime—if, for example, the rational number field Q is taken to include one “infinite prime” along with the finite primes 2, 3, 5, 7, and so on. A less vivid term for infinite primes is Archimedean places. For each ordinary prime number p the p-adic numbers Qp focus on the single factor p. From this point of view the real numbers R focus on absolute value, which at first glance is nothing like a prime factor, but there are extensive axiomatic analogies. Chevalley stressed how much simpler the theory becomes when “infinite primes” are put on a par with the finite.

The theory of prime factorization in any one local field is extremely simple because there is only one prime. The algebraic number fields K are global, as each one of these fields involves all the usual finite primes at once plus some infinite. Number theorists would calculate various class fields for K by quite complicated use of related local fields. Chevalley multiplied the number of basic definitions manyfold and yet simplified, unified, and extended the theory overall by introducing class fields directly for the local fields themselves. He earned his doctorate from the University of Paris in 1933 with a thesis on class field theory over finite fields and local fields.

The dissertation won Chevalley research support in Paris through 1937. For several reasons he worked on eliminating classical complex analysis from class field theory: the local fields related to prime numbers p favored

algebraic methods. Chevalley extended ideas from Wolfgang Krull and Herbrand to unify class field theory by generalizing algebraic number fields K to infinite degree extensions of Q (that is, extensions involving infinitely many independent algebraic irrationals) and these also favored algebraic tools. And Chevalley was personally drawn to modern, pure, uniform algebraic methods replacing classical complex analysis.

Chevalley advanced each of these goals by his creation of idèles linking local and global. Roughly speaking, an idèle of a global field K is a list of ways that a nonzero element of K might look in each of the local fields related to K. This list may or may not actually be generated by an element of K, just as an ideal of a ring may or may not be generated by a single ring element. So idèles are more flexible and more easily accessible than elements, and arithmetic conclusions about K follow from knowing which idèles correspond to elements. Weil added a related notion of adèle, and the two notions are today central to algebraic number theory.

Local fields and the Galois groups of infinite degree field extensions both have pro-finite topologies. For example, in the 5-adic integers Z5 a number counts as closer to 0 when it is divisible by a higher power of p (that is, when it equals 0 modulo a higher power). This was a triumph for the axiomatic definition of a topological space. All of these spaces satisfy the axioms. Topological notions of continuity, compactness, and so forth, are very useful in studying them. Yet they are far from ordinary spatial intuitions. For one thing, they are everywhere disconnected. The only connected parts of such a space are the single points, and yet these disconnected points “cluster around” one another. Chevalley applied the axioms completely unconcerned with classical geometric intuitions. Idèles also have an algebraically defined topology, although a version by Weil suited to Fourier analysis has replaced Chevalley’s version.

Bourbaki . Hadamard’s seminar ended in 1933. Gaston Julia allowed Weil, Chevalley, and other ENS graduates to run a new seminar in his name on recent mathematics ignored by the dominant mathematicians of Paris at the time. These meetings gave rise to a plan to replace the venerable Goursat (1902) by a new up-to-date textbook on analysis. And so Henri Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonné, René de Possel, and André Weil met on Monday 10 December 1934 at the Café Capoulade. The textbook project quickly expanded to a project for a book series on the basics of all pure mathematics, which would reestablish French dominance in mathematics. The group adopted the collective pseudonym of Nicolas Bourbaki, and the series became the phenomenally influential Elements of Mathematics. Chevalley’s mother hosted two early Bourbaki congresses at the family property at Chançay.

Dieudonné says that Chevalley was early a leader because “at the start only Weil and he had the vast mathematical culture required to conceive a plan for the whole, while each of the others only gradually acquired the necessary panoramic view” (1999, p. 107). Probably through Herbrand’s influence, Chevalley was an ardent advocate of Hilbert-school axiomatic rigor indifferent to any nonmathematical reality. This was common ground in Bourbaki, and Bourbaki had a large role in making it widespread today—despite complaints in some quarters that Bourbaki’s Elements promoted sterile abstraction. Chevalley outdid even Weil in this style, so that Weil reviewed Chevalley as “algebra with a vengeance; algebraic austerity could go no further …a valuable and useful book …[yet] severely dehumanized” (1951).

Chevalley did not see himself that way. He joined the personalist group “l’Ordre Nouveau,” not to be confused with a later extreme right group by the same name. They promoted personal liberty and growth and rejected all of anarchism, despotism, Marxism, and “le capitalism e sauvage,” or unrestrained capitalism.

Career . The single result most widely associated with Chevalley is the Chevalley-Warning theorem, far from his deepest theorem, and suggested by a less specific conjecture of Artin. The name credits Chevalley for seeing its importance as early as 1936. Any finite field has some prime number p as characteristic and then the number of elements is some power pn of p. Chevalley’s part of the theorem says: for any polynomial P(X1, …, Xj) with degree less than the number of variables, the number of roots in any finite field F is divisible by the characteristic of F. In particular, every homogeneous polynomial has a solution <0,…,0> so if the degree is less than the number of variables then it has nonzero solutions. This is important to projective geometry over finite fields.

Chevalley did his first teaching in 1936 at Strasbourg, replacing Weil, who had gone to the Institute for Advanced Study in Princeton, New Jersey, and then at Rennes in a teaching-research position (maître de conférences) 1937–1938, replacing Dieudonné who had gone to Nancy. In 1938 he was invited to the Institute for Advanced Study and he was there when World War II broke out in Europe. The French ambassador felt he might best serve France by remaining in the United States, where he was the only French scholar at the time.

Princeton University made him a professor, and he remained there until 1948, when he moved to Columbia University, where he stayed until 1955. American students found him terse and demanding, so that few undertook research with him. In 1933 Chevalley had married his first-cousin Jacqueline. There were no children, and the marriage dissolved in 1948. That same year he married his second wife, Sylvie, a professor and theater historian in New York. Their daughter Catherine was born in 1951 and became a philosopher and historian of science. When the family returned to France, Sylvie would become librarian and archivist for the Comédie Française.

In the 1940s Chevalley took up algebraic geometry and Lie groups and most especially the union of these themes in the Lie algebras of algebraic groups. He followed Weil in seeking algebraic geometry not only over the complex numbers but over an arbitrary field. In other words he wanted to eliminate classical complex analysis here too, in favor of purely algebraic methods. He would return to this in his 1950s Paris seminar, described below.

His greatest mathematical influence was in Lie groups and centered on his use of global topological methods regarding a Lie group as a manifold. Lie groups had always been defined as (real or complex) manifolds that are simultaneously groups, but workers from Sophus Lie (1842–1899) on had routine methods for looking at the group operation in an infinitesimal neighborhood of the unit element, while they looked at the space as a whole on a more ad hoc basis as needed. In precise terms, they had better tools for Lie algebras than for Lie groups directly.

The real line R is a Lie group with real-number addition as group law, and the unit circle S1 is a Lie group with angle addition as group law. A small neighborhood of the unit is the same in the two cases: In R it is 0 plus or minus some small real number. In S1 it is 0° plus or minus some small number of degrees. So R and S1 have the same Lie algebra. The difference is global: traveling away from 0 in R leads to ever new real numbers; while traveling away from 0° in S1 eventually circles back to 0°. Lie thought of his groups globally. Henri Poincaré (1854–1912) created his tools for topology largely to handle the global topology of Lie groups. Chevalley’s Princeton colleague Hermann Weyl made group representations central to his work, reflecting the global properties of groups. But no one before Chevalley succeeded at making global topology so explicitly fundamental to Lie group theory.

On the one hand this organized the theory so well that Chevalley (1946) became “the basic reference on Lie groups for at least two decades” (Dieudonné and Tits, 1987, p. 3). On the other hand it combined with Chevalley’s interest in finite fields and algebraic geometry. As the global theory was less analytic, it could generalize. An algebraic group is rather like a Lie group, possibly over some field other than the real or complex numbers. Each algebraic group has a kind of Lie algebra, but over most fields not all of these Lie algebras come from groups. Chevalley found a remarkable procedure whereby every simple Lie algebra (that is, one that is not abelian and contains no nontrivial ideals) over the complex numbers corresponds to a simple algebraic group over any field (that is, a group with no nontrivial quotients). A huge feat in itself, this fed into one of the largest projects in twentieth-century mathematics. Applied to any finite field it gives finite simple groups now called Chevalley groups; it was a useful organizing device, and gave some previously unknown finite simple groups. It became a tool in the vast and now completed project of describing all the finite simple groups. Further Chevalley hoped his algebraic group methods applied to algebraic number fields and related fields would be useful in arithmetic. Alexander Grothendieck’s algebraic geometry soon made them so.

A Guggenheim grant took Chevalley back to Paris for the year 1948–1949. He happily rejoined the Bourbaki circle and shared research with them. He was also happy to leave the anticommunist atmosphere of America during the 1948 Soviet blockade of Berlin, the 1949 Soviet atomic bomb test, and the imminent rise of McCarthyism. A Fulbright grant took him to Japan for 1953–1954, including three months at Nagoya. Chevalley had worked with Japanese mathematicians in Hamburg and later in Princeton and always maintained close ties to Japan.

Return to France . A 1954 campaign to place Chevalley at the Sorbonne “unleashed passions in the mathematical community scarcely comprehensible today” (Dieudonné, 1999, p. 110). Some believed new opportunities should go to those who had stayed and fought or been captured in the army or the resistance. Others opposed Bourbaki’s influence. Chevalley was appointed in 1955 to the nearby Université de Paris VII, where he retired in 1978.

He began a seminar in 1956 at the École Normale Supérieur, initially with Cartan, and made it an early home for Grothendieck’s project to rewrite the foundations of algebraic geometry. Chevalley among others anticipated aspects of Grothendieck’s scheme theory, and the seminar proceedings include the first published use of the word scheme (schéma) in its current sense in algebraic geometry. He claimed “Grothendieck had advanced algebraic geometry by fifty years” (Seshadri, 1999, p. 120).

Chevalley continued to deepen his work on groups. He determined all of the semisimple groups over any algebraically closed field, that is, all of the groups that cannot be reduced in a certain way to simple groups. When the field has characteristic 0, or in other words contains a copy of the rational numbers Q, both the result and the proof were very like those known fifty years before over the complex numbers using methods of Lie algebras. But when the field has a finite prime characteristic p, so that it contains a copy of the finite field of p elements, the older proof did not work at all. Lie algebras in finite characteristic were not fully classified themselves, they were known to be much more complicated than in characteristic 0, and Chevalley found they tell much less about the Lie groups. Surprisingly, then, when Chevalley completed the proof for finite characteristic the result was independent of which finite characteristic, and rather parallel to characteristic 0. Since Lie algebra methods were unusable, he produced purely group theoretic and algebra-geometric methods and the deepest work of his career. In the 1960s he turned to more detailed work on finite groups and produced and inspired new developments on them.

He retained his philosophical and political ideas though he had lost the friends who shared them; the last of his philosophical friends, Albert Lautman, died in the Resistance. Having become skilled at the game of Go in Japan, Chevalley promoted it among his friends and students, some of whom organized the first Go club in France in 1969. He supported the student movement of 1968 and charitable progressive projects of a kind associated with French Protestantism though he did not keep the religious faith. Chevalley had small respect for academic honors. He accepted the Cole Prize of the American Mathematical Society in 1941, and honorary membership in the London Mathematical Society in 1967. Dieudonné wrote, “No one who has known Chevalley can fail to see how his [philosophical-political] principles agreed with his entire character. When his articles exalt ‘revulsion at accommodations, at self-satisfaction or satisfaction with humanity in general, and at every kind of hypocrisy’ he describes himself” (1999, p. 112).

BIBLIOGRAPHY

WORKS BY CHEVALLEY

“Sur la théorie des idéaux dans les corps algébriques infini.” Academie des Sciences, Comptes Rendus Hebdomadaires 189 (1929): 616–618.

With Arnaud Dandieu. “Logique hilbertienne et psychologie.” Revue philosophique de la France et de l’Étranger 113 (1932): 99–111. Depicts Hilbert’s proof theory as an apt reaction to Bertrand Russell’s realist logicism on one hand and Luitzen Egbertus Jan Brouwer’s intuitionism on the other.

“Démonstration d’une hypothèse de M. Artin.” Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg 11 (1936): 73–75.

“La théorie du corps de classes.” Annals of Mathematics, ser. 2, 41 (1940): 394–418.

Theory of Lie Groups. Vol. 1. Princeton, NJ: Princeton University Press, 1946.

Introduction to the Theory of Algebraic Functions of One Variable. New York: American Mathematical Society, 1951.

Théorie des groupes de Lie. Vol. 2, Groupes algébriques. Paris: Hermann, 1951.

Théorie des groupes de Lie. Vol. 3, Théorèmes généraux sur les algèbres de Lie. Paris: Hermann, 1955.

Fondaments de la géométrie algébrique. Paris: Secrétariat Mathématique, 1958.

OTHER SOURCES

Dieudonné, Jean. “Claude Chevalley.” Transformation Groups 4, nos. 2–3 (1999): 105–118. This includes an extensive bibliography compiled by Catherine Chevalley.

Dieudonné, Jean, and Jacques Tits. “Claude Chevalley (1909–1984).” Bulletin of the American Mathematical Society17 (1987): 1–7. A masterful mathematical survey.

Goursat, Édouard. Cours d’analyse mathématique. 2 vols. Paris: Gauthier-Villars, 1902–1905.

Guedj, Denis. “Nicholas Bourbaki, Collective Mathematician: An Interview with Claude Chevalley.” Mathematical Intelligencer 7, no. 2 (1985): 18–22.

Hasse, Helmut. “History of Class Field Theory.” In Algebraic Number Theory: Proceedings of an Instructional Conference Organized by the London Mathematical Society (a NATO Advanced Study Institute) with the Support of the International Mathematical Union, edited by John Cassels and Albrecht Fröhlich, 266–279. London: Academic Press, 1967.

Seshadri, Conjeevaram. “Claude Chevalley: Some Reminiscences.” Transformation Groups 4, nos. 2–3 (1999): 119–125.

Weil, André. Review of Introduction to the Theory of Algebraic Functions of One Variable, by C. Chevalley. Bulletin of the American Mathematical Society 57 (1951): 384–398.

Colin McLarty

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