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Adele ring
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=== Motivation === The ring of adeles solves the technical problem of "doing analysis on the rational numbers <math>\mathbf{Q}</math>." The classical solution was to pass to the standard metric completion <math>\mathbf{R}</math> and use analytic techniques there.{{What|date=May 2023|reason=pass what? use what analytical techniques?}} But, as was learned later on, there are many more [[Absolute value (algebra)|absolute values]] other than the [[Euclidean distance]], one for each prime number <math>p \in \mathbf{Z}</math>, as classified by [[Ostrowski's theorem]]. The Euclidean absolute value, denoted <math>|\cdot|_\infty</math>, is only one among many others, <math>|\cdot |_p</math>, but the ring of adeles makes it possible to comprehend and {{Em|use all of the valuations at once}}. This has the advantage of enabling analytic techniques while also retaining information about the primes, since their structure is embedded by the restricted infinite product. The purpose of the adele ring is to look at all completions of <math>K</math> at once. The adele ring is defined with the restricted product, rather than the [[Cartesian product]]. There are two reasons for this: * For each element of <math>K</math> 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. ==== 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>
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