Claude Chevalley

Claude Chevalley (11 February 1909 - 28 June 1984) was a French mathematician with an austere style based on abstract algebra. He was born in Johannesburg, and died in Paris. He was a founder member of the Bourbaki group.

After graduating from the École Normale Supérieur, he studied under Emil Artin in Hamburg from 1931, a time which may have formed his mathematical taste, and then under Helmut Hasse. One of his achievements was a step in the technical development of class field theory, removing a use of L-functions and replacing it by an algebraic method. At that time use of group cohomology was implicit, cloaked by the language of central simple algebras. In the introduction to his book Basic Number Theory, Chevalley's friend André Weil explains that the book's adoption of that road goes back to an old, unpublished manuscript of Chevalley.

Chevalley also wrote a three-volume treatment of Lie groups around 1950. A few years later he published an investigation into what are now called Chevalley groups, and for which he is most remembered. An accurate discussion of conditions of integrality in the Lie algebras of semisimple groups enabled their theory to be abstracted from the real and complex fields. As a consequence, analogues over finite fields could be defined. This was an essential stage in the theory of the finite simple groups. After Chevalley's work the distinction between 'classical groups' falling into the Dynkin diagram classification, and 'sporadic 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 usually refers to his result on solubility of equations over a finite field (also called the Chevalley-Warning theorem). Another theorem concerns the constructible sets in algebraic geometry: those in the Boolean algebra generated by the Zariski-open and closed sets. It states that the image of such a set by a morphism of algebraic varieties is of the same type. In logicians' terms, this is an 'elimination of quantifiers'.