Adele Ring

In algebraic number theory and topological algebra, the adele ring (other names are the adelic ring, the ring of adeles) is a self dual topological ring which is built on the field of rational numbers (or, more generally, any algebraic number field). It involves in a symmetric way all the completions of the field.

The adele ring was invented by Claude Chevalley for the purposes of simplifying and clarifying class field theory. It has also found applications outside that area.

The adele ring and its relation to the number field are among the most fundamental objects in number theory. The quotient of its multiplicative group by the multiplicative group of the algebraic number field is the central object in class field theory. It is a central principle of Diophantine geometry to study solutions of polynomials equations in number fields by looking at their solutions in the larger complete adele ring, where it is generally easier to detect solutions, and then deciding which of them come from the number field.

The word "adele" is short for "additive idele" and it was invented by André Weil. The previous name was the valuation vectors. The ring of adeles was historically preceded by the ring of repartitions, a construction which avoids completions, and is today sometimes referred to as pre-adele.