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Adele ring
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====Characterisation of the idele group==== :'''Theorem.'''<ref>The general proof of this theorem for any global field is given in {{harvnb|Weil|1967|p=77.}}</ref> Let <math>K</math> be a number field. There exists a finite set of places <math>S,</math> such that: ::<math>I_K= \left (I_{K,S} \times \prod_{v \notin S} O_v^\times \right ) K^\times= \left(\prod_{v \in S} K_v^\times \times \prod_{v \notin S} O_v^\times\right) K^\times.</math> '''Proof.''' The [[Ideal class group|class number]] of a number field is finite so let <math>\mathfrak{a}_1, \ldots, \mathfrak{a}_h</math> be the ideals, representing the classes in <math>\operatorname{Cl}_K.</math> These ideals are generated by a finite number of prime ideals <math>\mathfrak{p}_1, \ldots, \mathfrak{p}_n.</math> Let <math>S</math> be a finite set of places containing <math>P_\infty</math> and the finite places corresponding to <math>\mathfrak{p}_1, \ldots, \mathfrak{p}_n.</math> Consider the isomorphism: :<math>I_K/ \left(\prod_{v< \infty}O_v^\times \times \prod_{v | \infty}K_v^\times\right) \cong J_K,</math> induced by :<math>(\alpha_v)_v \mapsto \prod_{v < \infty} \mathfrak{p}_v^{v(\alpha_v)}.</math> At infinite places the statement is immediate, so the statement has been proved for finite places. The inclusion β³<math>\supset</math>β³ is obvious. Let <math>\alpha \in I_{K,\text{fin}}.</math> The corresponding ideal <math>\textstyle (\alpha)=\prod_{v< \infty} \mathfrak{p}_v^{v(\alpha_v)}</math> belongs to a class <math>\mathfrak{a}_iK^{\times},</math> meaning <math>(\alpha)=\mathfrak{a}_i(a)</math> for a principal ideal <math>(a).</math> The idele <math>\alpha'=\alpha a^{-1}</math> maps to the ideal <math>\mathfrak{a}_i</math> under the map <math>I_{K,\text{fin}} \to J_K.</math> That means <math>\textstyle \mathfrak{a}_i=\prod_{v< \infty} \mathfrak{p}_v^{v(\alpha'_v)}.</math> Since the prime ideals in <math>\mathfrak{a}_i</math> are in <math>S,</math> it follows <math>v(\alpha'_v)=0</math> for all <math>v \notin S,</math> that means <math>\alpha'_v \in O_v^\times</math> for all <math>v \notin S.</math> It follows, that <math>\alpha'=\alpha a^{-1} \in I_{K,S},</math> therefore <math>\alpha \in I_{K,S}K^\times.</math>
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