Adele ring

In mathematics the adele ring is defined in class field theory, a branch of (algebraic) number theory. It allows one to elegantly describe the Artin reciprocity law. The adele ring is a self-dual topological ring, which is built on a global field. It is the restricted product of all the completions of the global field and therefore contains all the completions of the global field.

The idele class group, which is the quotient group of the group of units of the adele ring by the group of units of the global field, is a central object in class field theory.

Notation: During the whole article, ${\displaystyle K}$ is a global field. That means, that ${\displaystyle K}$ is an algebraic number field or a global function field. In the first case, ${\displaystyle K/\mathbb {Q} }$ is a finite field extension, in the second case ${\displaystyle K/\mathbb {F} _{p^{r}}(t)}$ is a finite field extension. We write ${\displaystyle v}$ for a place of ${\displaystyle K,}$ that means ${\displaystyle v}$ is a representative of an equivalence class of . The trivial valuation and the corresponding trivial value aren't allowed in the whole article. A finite/non-Archimedean valuation is written as ${\displaystyle v<\infty }$ or ${\displaystyle v\nmid \infty }$ and an infinite/Archimedean valuation as ${\displaystyle v\mid \infty .}$ We write ${\displaystyle P_{\infty }}$ for the finite set of all infinite places of ${\displaystyle K}$ and ${\displaystyle P}$ for a finite subset of all places of ${\displaystyle K,}$ which contains ${\displaystyle P_{\infty }.}$ In addition, we write ${\displaystyle K_{v}}$ for the completion of ${\displaystyle K}$ with respect to the valuation ${\displaystyle v.}$ If the valuation ${\displaystyle v}$ is discrete, then we write ${\displaystyle {\mathcal {O}}_{v}}$ for the valuation ring of ${\displaystyle K_{v}.}$ We write ${\displaystyle {\mathfrak {m}}_{v}}$ for the maximal ideal of ${\displaystyle {\mathcal {O}}_{v}.}$ If this is a principal ideal, then we write ${\displaystyle \pi _{v}}$ for a uniformizing element. By fixing a suitable constant ${\displaystyle C>1,}$ there is a one-to-one identification of valuations and absolute values: The valuation ${\displaystyle v}$ is assigned the absolute value ${\displaystyle |\cdot |_{v},}$ which is defined as:

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