Claude Chevalley (French:[ʃəvalɛ]; 11 February 1909 – 28 June 1984) was a French mathematician who made important contributions tonumber theory,algebraic geometry,class field theory,finite group theory and the theory ofalgebraic groups. He was a founding member of theBourbaki group.

Claude Chevalley
Y. Akizuki, C. Chevalley and A. Kobori
Born	(1909-02-11)February 11, 1909 Johannesburg,Transvaal Colony (now South Africa)
Died	June 28, 1984(1984-06-28) (aged 75) Paris, France
Nationality	French
Citizenship	French, American
Alma mater	École Normale Supérieure University of Hamburg University of Marburg University of Paris
Known for	Founding member ofBourbaki Chevalley–Warning theorem Chevalley group Chevalley scheme
Scientific career
Fields	Mathematics
Institutions	Princeton University Columbia University
Notable students	Michel André Michel Broué Leon Ehrenpreis Oscar Goldman Gerhard Hochschild Lê Dũng Tráng

His father, Abel Chevalley, was a French diplomat who, jointly with his wife Marguerite Chevalley néeSabatier, wroteThe Concise Oxford French Dictionary.^[1] Chevalley graduated from theÉcole Normale Supérieure in 1929, where he studied underÉmile Picard. He then spent time at theUniversity of Hamburg, studying underEmil Artin and at theUniversity of Marburg, studying underHelmut Hasse. In Germany, Chevalley discovered Japanese mathematics in the person ofShokichi Iyanaga. Chevalley was awarded a doctorate in 1933 from theUniversity of Paris for a thesis onclass field theory.

When World War II broke out, Chevalley was atPrinceton University. After reporting to the French Embassy, he stayed in the U.S., first at Princeton and then (after 1947) atColumbia University. His American students includedLeon Ehrenpreis andGerhard Hochschild. During his time in the U.S., Chevalley became an American citizen and wrote a substantial part of his lifetime's output in English.

When Chevalley applied for a chair at theSorbonne, the difficulties he encountered were the subject of a polemical piece by his friend and fellowBourbakisteAndré Weil, titled "Science Française?" and published in theNouvelle Revue Française. Chevalley was the "professeur B" of the piece, as confirmed in the endnote to the reprint in Weil's collected works,Oeuvres Scientifiques, tome II. Chevalley eventually did obtain a position in 1957 at the faculty of sciences of theUniversity of Paris and after 1970 at theUniversité de Paris VII.

Chevalley had artistic and political interests, and was a minor member of the Frenchnon-conformists of the 1930s. The following quote by the co-editor of Chevalley's collected works attests to these interests:

"Chevalley was a member of various avant-garde groups, both in politics and in the arts... 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."^[2]

In his PhD thesis, Chevalley made an important contribution to the technical development ofclass field theory, removing a use ofL-functions and replacing it by an algebraic method. At that time use ofgroup cohomology was implicit, cloaked by the language ofcentral simple algebras. In the introduction toAndré Weil'sBasic Number Theory, Weil attributed the book's adoption of that path to an unpublished manuscript by Chevalley.

Around 1950, Chevalley wrote a three-volume treatment ofLie groups. A few years later, he published the work for which he is best remembered, his investigation into what are now calledChevalley groups. Chevalley groups make up 9 of the 18 families offinite simple groups.

Chevalley's accurate discussion of integrality conditions in theLie algebras ofsemisimple groups enabled abstracting their theory from thereal andcomplex fields. As a consequence, analogues overfinite fields could be defined. This was an essential stage in the evolvingclassification of finite simple groups. After Chevalley's work, the distinction between "classical groups" falling into theDynkin diagram classification, andsporadic groups which did not, became sharp enough to be useful. What are called 'twisted' groups of the classical families could be fitted into the picture.

"Chevalley's theorem" (also called theChevalley–Warning theorem) usually refers to his result on the solubility of equations over a finite field. Another theorem of his concerns theconstructible sets inalgebraic geometry, i.e. those in theBoolean algebra generated by theZariski-open andZariski-closed sets. It states that theimage of such a set by amorphism ofalgebraic varieties is of the same type. Logicians call this anelimination of quantifiers.

In the 1950s, Chevalley led some Paris seminars of major importance: theSéminaire Cartan–Chevalley of the academic year 1955-6, withHenri Cartan and theSéminaire Chevalley of 1956-7 and 1957-8. These dealt with topics onalgebraic groups and the foundations of algebraic geometry, as well as pureabstract algebra. The Cartan–Chevalley seminar was the genesis ofscheme theory, but its subsequent development in the hands ofAlexander Grothendieck was so rapid, thorough and inclusive that its historical tracks can appear well covered. Grothendieck's work subsumed the more specialised contribution ofSerre, Chevalley,Gorō Shimura and others such asErich Kähler andMasayoshi Nagata.

The American Mathematical Society established the Chevalley Prize in Lie Theory^[3] in 2014, with the first recipient beingGeordie Williamson in 2016.

• 1936.L'Arithmetique dans les Algèbres de Matrices. Hermann, Paris.^[4]
• 1940. "La théorie du corps de classes,"Annals of Mathematics 41: 394–418.
• 1946.Theory of Lie groups. Princeton University Press.^[5]
• 1951."Théorie des groupes de Lie, tome II, Groupes algébriques", Hermann, Paris.
• 1951.Introduction to the theory of algebraic functions of one variable, A.M.S. Math. Surveys VI.^[6]
• 1954.The algebraic theory of spinors, Columbia Univ. Press;^[7] new edition, Springer-Verlag, 1997.
• 1953–1954.Class field theory, Nagoya University.
• 1955."Théorie des groupes de Lie, tome III, Théorèmes généraux sur les algèbres de Lie", Hermann, Paris.
• 1955, "Sur certains groupes simples,"Tôhoku Mathematical Journal 7: 14–66.
• 1955.The construction and study of certain important algebras, Publ. Math. Soc. Japan.^[8]
• 1956.Fundamental concepts of algebra, Acad. Press.^[9]
• 1956–1958. "Classification des groupes de Lie algébriques", Séminaire Chevalley, Secrétariat Math., 11 rue P. Curie, Paris; revised edition by P.Cartier, Springer-Verlag, 2005.
• 1958.Fondements de la géométrie algébrique, Secrétariat Math., 11 rue P. Curie, Paris.

• Idèle
• Valuative criterion of properness
• Chevalley group
• Chevalley scheme
• Chevalley–Iwahori–Nagata theorem
• Beck–Chevalley condition
• Non-conformist movement
• Jordan–Chevalley decomposition

• O'Connor, John J.;Robertson, Edmund F.,"Claude Chevalley",MacTutor History of Mathematics Archive,University of St Andrews
• Claude Chevalley at theMathematics Genealogy Project