The role held in local class field theory by the multiplicative group of the base field is taken in global class field theory by the idele class group. The notion of idele is a modification of the notion of ideal. It was introduced by the French mathematician Claude Chevalley (1909–1984) with a view to providing a suitable basis for the important local-to-global principle, i.e., for the principle which reduces problems concerning a number field K to analogous problems for the various completions K p . Chevalley used the term “ideal element”, which was abbreviated as id. el.