structure inside of , making it possible to build a theory of Fourier analysis (cf. Harmonic analysis) in the adelic setting. This is analogous to the situation in algebraic number theory where the ring of integers of an algebraic number field embedsas a lattice. With the power of a new theory of Fourier analysis, Tate was able to prove a special class of L-functions and the Dedekind zeta functions were meromorphic on the complex plane. Another natural reason for why this technical condition holds can be seen directly by constructing the ring of adeles as a tensor product of rings. If defining the ring of integral adeles as the ringthen the ring of adeles can be equivalently defined asThe restricted product structure becomes transparent after looking at explicit elements in this ring. If taking a rational number it is found . For any tuple there are the following series of equalitiesThen, for any there is for , but for since there is an inverse power of . This shows any element in this new ring of adeles can have an element in at only finitely many places . Origin of the name In local class field theory, the group of units of the local field plays a central role. In global class field theory, the idele class group takes this role. The term "idele" () is an invention of the French mathematician Claude Chevalley (1909–1984) and stands for "ideal element" (abbreviated: id.el.). The term "adele" () stands for additive idele. The idea of the adele ring is to look at all completions of at once. At first glance, the Cartesian product could be a good candidate. However, the adele ring is defined with the restricted product. There are two reasons for this: For each element of the valuations are zero for almost all places, i.e., for all places except a finite number. So, the global field can be embedded in the restricted product. The restricted product is a locally compact space, while the Cartesian product is not. Therefore, there cannot be any application of harmonic analysis to the Cartesian product. This is because local compactness ensures the existence (and uniqueness) of Haar measure, a crucial tool in analysis on groups in general. Examples Ring of adeles for the rational numbers The rationals K=Q have a valuation for every prime number p, with (Kν,Oν)=(Qp,Zp), and one infinite valuation ∞ with Q∞=R.