Idèle

An invertible element of the ring of adèles (cf. Adèle). The set of all idèles forms a group under multiplication, called the idèle group. The elements of the idèle group of the field of rational numbers are sequences of the form

where is a non-zero real number, is a non-zero -adic number, and for all but finitely many (here is the -adic norm). A sequence of idèles

is said to converge to an idèle if it converges to componentwise and if there exists an such that for and all . The idèle group is a locally compact topological group in this topology. The idèle group of an arbitrary number field is constructed in an analogous way.

The multiplicative group of the field of rational numbers is isomorphically imbedded in the idèle group of this field. Every rational number is associated with the sequence

which is an idèle. Such an idèle is said to be a principal idèle. The subgroup consisting of all principal idèles is a discrete subgroup of the idèle group.

The concepts of an idèle and an adèle were introduced by C. Chevalley in 1936 for the purposes of algebraic number theory. The new language proved useful in the study of arithmetic aspects of the theory of algebraic groups. To those ends, A. Weil generalized the definitions of an adèle and an idèle to the case of an arbitrary linear algebraic group defined over a number field.

References

[1]	A. Weil, "Basic number theory" , Springer (1973)
[2]	J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986)

Comments

Let be an index set and for each let there be given a locally compact topological ring or group and an open compact subring or subgroup . The restricted direct product of the with respect to the consists of all families such that for all but finitely many . becomes a locally compact group (ring) by taking as a basis of open neighbourhoods of the identity (zero) the sets with open in for all and for all but finitely many . For each finite set let . Then is the union (direct limit) of the .

Now let be a number field (or, more generally, a global field). Let be the set of all prime divisors of (both finite and infinite ones). For each let be the completion of with respect to the norm of , and let be the ring of integers of . (Set if is infinite.) Then the restricted product of the with respect to the is the ring of adèles of .

Now for each let be the group of non-zero elements of and let be the group of units of (if is infinite take ). The restricted product of the with respect to the is the group of idèles of . As a set the group of idèles is the set of invertible elements of . But the topology on is stronger than that induced by .

The quotient of by the diagonal subgroup of principal idèles is called the idèle class group; it is important in class field theory.

The name idèle derives from ideal element. This got abbreviated id.el., which, pronounced in French, gave rise to idèle.