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Claude Chevalley was the only son of Abel and Marguerite Chevalley who were the authors of the Oxford Concise French Dictionary. He studied under Emile Picard at the Ecole Normale Supérieur in Paris graduating in 1929. He went on to become the youngest of the Bourbaki mathematicians.
After graduating Chevalley continued his studies in Germany, studying under Artin at Hamburg during session 193132. 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 in 1937.
In 1938 Chevalley went to the United States to the Institute of Advanced Study at Princeton where he also served on the Faculty of the University of Princeton. From July 1949 until June 1957 he served as professor of mathematics at Columbia University, afterwards returning to the University of Paris.
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 quasialgebraically 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 semisimple algebraic group.
Many of his texts have become classics. He wrote Theory of Lie Groups in three volumes which appeared in 1946, 1951 and 1955. He 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), Fundamental concepts of algebra (1956) and Foundations of algebraic geometry (1958).
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.
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Claude Chevalley was the American Mathematical Society Cole Prize winner in 1941. You can see a history of the AMS Cole Prize and a list of the winners.