Editing Adele ring (section)

==== Why the restricted product? ====
The [[Restricted product|restricted infinite product]] is a required technical condition for giving the number field <math>\mathbf{Q}</math> a lattice structure inside of <math>\mathbf{A}_\mathbf{Q}</math>, 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 embeds<blockquote><math>\mathcal{O}_K \hookrightarrow K</math></blockquote>as a lattice. With the power of a new theory of Fourier analysis, [[John Tate (mathematician)|Tate]] was able to prove a special class of [[L-function]]s and the [[Dedekind zeta function]]s were [[Meromorphic function|meromorphic]] on the complex plane. Another natural reason for why this technical condition holds can be seen by constructing the ring of adeles as a tensor product of rings. If defining the ring of integral adeles <math>\mathbf{A}_\mathbf{Z}</math> as the ring<blockquote><math>\mathbf{A}_\mathbf{Z} = \mathbf{R}\times\hat{\mathbf{Z}} = \mathbf{R}\times \prod_p \mathbf{Z}_p,</math></blockquote>then the ring of adeles can be equivalently defined as<blockquote><math>\begin{align}
\mathbf{A}_\mathbf{Q} &= \mathbf{Q}\otimes_\mathbf{Z}\mathbf{A}_\mathbf{Z} \\
&= \mathbf{Q}\otimes_\mathbf{Z} \left( \mathbf{R}\times \prod_{p} \mathbf{Z}_p \right).
\end{align}</math></blockquote>The restricted product structure becomes transparent after looking at explicit elements in this ring. The image of an element <math>b/c\otimes(r,(a_p)) \in \mathbf{A}_\mathbf{Q}</math> inside of the unrestricted product <math display=inline> \mathbf{R}\times \prod_p \mathbf{Q}_p</math> is the element <blockquote><math>
\left(\frac{br}{c}, \left(\frac{ba_p}{c}\right) \right). </math></blockquote> The factor <math>ba_p/c</math> lies in <math>\mathbf{Z}_p</math> whenever <math>p</math> is not a prime factor of <math>c</math>, which is the case for all but finitely many primes <math>p</math>.<ref>{{Cite web|url=https://ncatlab.org/nlab/show/ring+of+adeles|title=ring of adeles in nLab|website=ncatlab.org}}</ref>