adele ring

{{More citations needed section|date=May 2023}}

Let $K$ be a global field (a finite extension of $\backslash mathbf\{Q\}$ or the function field of a curve $X/\backslash mathbf\{F\_\{\backslash mathit\{q\}\}\}$ over a finite field). The adele ring of $K$ is the subring

:$\backslash mathbf\{A\}\_K\backslash \; =\; \backslash \; \backslash prod\; (K\_\backslash nu,\backslash mathcal\{O\}\_\backslash nu)\; \backslash \; \backslash subseteq\; \backslash \; \backslash prod\; K\_\backslash nu$

consisting of the tuples $(a\_\backslash nu)$ where $a\_\backslash nu$ lies in the subring $\backslash mathcal\{O\}\_\backslash nu\; \backslash subset\; K\_\backslash nu$ for all but finitely many places $\backslash nu$. Here the index $\backslash nu$ ranges over all valuations of the global field $K$, $K\_\backslash nu$ is the completion at that valuation and $\backslash mathcal\{O\}\_\backslash nu$ the corresponding valuation ring.{{Cite book |last=Sutherland |first=Andrew |url=https://math.mit.edu/classes/18.785/2015fa/LectureNotes22.pdf |title=18.785 Number theory I Lecture #22 |date=1 December 2015 |publisher=MIT |pages=4}}

The ring of adeles solves the technical problem of "doing analysis on the rational numbers $\backslash mathbf\{Q\}$." The classical solution was to pass to the standard metric completion $\backslash mathbf\{R\}$ 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 values other than the Euclidean distance, one for each prime number $p\; \backslash in\; \backslash mathbf\{Z\}$, as classified by Ostrowski's theorem. The Euclidean absolute value, denoted $|\backslash cdot|\_\backslash infty$, is only one among many others, $|\backslash cdot\; |\_p$, 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 $K$ 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 $K$ 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.

The restricted infinite product is a required technical condition for giving the number field $\backslash mathbf\{Q\}$ a lattice structure inside of $\backslash mathbf\{A\}\_\backslash mathbf\{Q\}$, 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

$\backslash mathcal\{O\}\_K\; \backslash hookrightarrow\; K$

$\backslash mathbf\{A\}\_\backslash mathbf\{Z\}\; =\; \backslash mathbf\{R\}\backslash times\backslash hat\{\backslash mathbf\{Z\}\}\; =\; \backslash mathbf\{R\}\backslash times\; \backslash prod\_p\; \backslash mathbf\{Z\}\_p,$

$\backslash begin\{align\}$

\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}

$$

\left(\frac{br}{c}, \left(\frac{ba_p}{c}\right) \right).

The term "idele" ({{langx|fr|idèle}}) is an invention of the French mathematician Claude Chevalley (1909–1984) and stands for "ideal element" (abbreviated: id.el.). The term "adele" (French: {{lang|fr|adèle}}) stands for additive idele. Thus, an adele is an additive ideal element.

The rationals $K\; =\; \backslash bold\{Q\}$ have a valuation for every prime number $p$, with $(\; K\_\backslash nu,\backslash mathcal\{O\}\_\backslash nu)=(\backslash mathbf\{Q\}\_p,\backslash mathbf\{Z\}\_p)$, and one infinite valuation ∞ with $\backslash mathbf\{Q\}\_\backslash infty\; =\; \backslash mathbf\{R\}$. Thus an element of

:$\backslash mathbf\{A\}\_\backslash mathbf\{Q\}\backslash \; =\; \backslash \; \backslash mathbf\{R\}\backslash times\; \backslash prod\_p\; (\backslash mathbf\{Q\}\_p,\backslash mathbf\{Z\}\_p)$

is a real number along with a p-adic rational for each $p$ of which all but finitely many are p-adic integers.

Secondly, take the function field $K=\backslash mathbf\{F\}\_q(\backslash mathbf\{P\}^1)=\backslash mathbf\{F\}\_q(t)$ of the projective line over a finite field. Its valuations correspond to points $x$ of $X=\backslash mathbf\{P\}^1$, i.e. maps over $\backslash text\{Spec\}\backslash mathbf\{F\}\_\{q\}$

:$x\backslash \; :\backslash \; \backslash text\{Spec\}\backslash mathbf\{F\}\_\{q^n\}\backslash \; \backslash longrightarrow\; \backslash \; \backslash mathbf\{P\}^1.$

For instance, there are $q+1$ points of the form $\backslash text\{Spec\}\backslash mathbf\{F\}\_\{q\}\backslash \; \backslash longrightarrow\; \backslash \; \backslash mathbf\{P\}^1$. In this case $\backslash mathcal\{O\}\_\backslash nu=\backslash widehat\{\backslash mathcal\{O\}\}\_\{X,x\}$ is the completed stalk of the structure sheaf at $x$ (i.e. functions on a formal neighbourhood of $x$) and $K\_\backslash nu=K\_\{X,x\}$ is its fraction field. Thus

:$\backslash mathbf\{A\}\_\{\backslash mathbf\{F\}\_q(\backslash mathbf\{P\}^1)\}\backslash \; =\backslash \; \backslash prod\_\{x\backslash in\; X\}\; (\backslash mathcal\{K\}\_\{X,x\},\backslash widehat\{\backslash mathcal\{O\}\}\_\{X,x\}).$

The same holds for any smooth proper curve $X/\backslash mathbf\{F\_\{\backslash mathit\{q\}\}\}$ over a finite field, the restricted product being over all points of $x\; \backslash in\; X$.

The group of units in the adele ring is called the idele group

:$I\_K\; =\; \backslash mathbf\{A\}\_\{K\}^\backslash times$.

The quotient of the ideles by the subgroup $K^\backslash times\; \backslash subseteq\; I\_K$ is called the idele class group

:$C\_K\backslash \; =\backslash \; I\_K/K^\backslash times.$

The integral adeles are the subring

:$\backslash mathbf\{O\}\_K\backslash \; =\backslash \; \backslash prod\; O\_\backslash nu\; \backslash \; \backslash subseteq\; \backslash \; \backslash mathbf\{A\}\_K.$

The Artin reciprocity law says that for a global field $K$,

:$\backslash widehat\{C\_K\}\; =\; \backslash widehat\{\backslash mathbf\{A\}\_K^\backslash times/K^\backslash times\}\; \backslash \; \backslash simeq\; \backslash \; \backslash text\{Gal\}(K^\backslash text\{ab\}/K)$

where $K^\{ab\}$ is the maximal abelian algebraic extension of $K$ and $\backslash widehat\{(\backslash dots)\}$ means the profinite completion of the group.

If $X/\backslash mathbf\{F\_\{\backslash mathit\{q\}\}\}$ is a smooth proper curve then its Picard group is{{Citation |title=Geometric Class Field Theory, notes by Tony Feng of a lecture of Bhargav Bhatt |url=https://math.berkeley.edu/~fengt/2GeometricCFT.pdf}}.

:$\backslash text\{Pic\}(X)\; \backslash \; =\; \backslash \; K^\backslash times\backslash backslash\; \backslash mathbf\{A\}^\backslash times\_X/\backslash mathbf\{O\}\_X^\backslash times$

and its divisor group is $\backslash text\{Div\}(X)=\backslash mathbf\{A\}^\backslash times\_X/\backslash mathbf\{O\}\_X^\backslash times$. Similarly, if $G$ is a semisimple algebraic group (e.g. $SL\_n$, it also holds for $GL\_n$) then Weil uniformisation says that{{Citation | title=Weil uniformization theorem, nlab article | url=https://ncatlab.org/nlab/show/Weil+uniformization+theorem}}.

:$\backslash text\{Bun\}\_G(X)\; \backslash \; =\; \backslash \; G(K)\backslash backslash\; G(\backslash mathbf\{A\}\_X)/G(\backslash mathbf\{O\}\_X).$

Applying this to $G=\backslash mathbf\{G\}\_m$ gives the result on the Picard group.

There is a topology on $\backslash mathbf\{A\}\_K$ for which the quotient $\backslash mathbf\{A\}\_K/K$ is compact, allowing one to do harmonic analysis on it. John Tate in his thesis "Fourier analysis in number fields and Hecke Zeta functions"{{sfn|Cassels|Fröhlich|1967}} proved results about Dirichlet L-functions using Fourier analysis on the adele ring and the idele group. Therefore, the adele ring and the idele group have been applied to study the Riemann zeta function and more general zeta functions and the L-functions.

If $X$ is a smooth proper curve over the complex numbers, one can define the adeles of its function field $\backslash mathbf\{C\}(X)$ exactly as the finite fields case. John Tate proved{{Citation | title=Residues of differentials on curves | year=1968| doi=10.24033/asens.1162| url=http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1968_4_1_1/ASENS_1968_4_1_1_149_0/ASENS_1968_4_1_1_149_0.pdf| last1=Tate| first1=John| journal=Annales Scientifiques de l'École Normale Supérieure| volume=1| pages=149–159}}. that Serre duality on $X$

:$H^1(X,\backslash mathcal\{L\})\backslash \; \backslash simeq\; \backslash \; H^0(X,\backslash Omega\_X\backslash otimes\backslash mathcal\{L\}^\{-1\})^*$

can be deduced by working with this adele ring $\backslash mathbf\{A\}\_\{\backslash mathbf\{C\}(X)\}$. Here L is a line bundle on $X$.

Throughout this article, $K$ is a global field, meaning it is either a number field (a finite extension of $\backslash Q$) or a global function field (a finite extension of $\backslash mathbb\{F\}\_\{p^r\}(t)$ for $p$ prime and $r\; \backslash in\; \backslash N$). By definition a finite extension of a global field is itself a global field.

For a valuation $v$ of $K$ it can be written $K\_v$ for the completion of $K$ with respect to $v.$ If $v$ is discrete it can be written $O\_v$ for the valuation ring of $K\_v$ and $\backslash mathfrak\{m\}\_v$ for the maximal ideal of $O\_v.$ If this is a principal ideal denoting the uniformising element by $\backslash pi\_v.$ A non-Archimedean valuation is written as or $v\; \backslash nmid\; \backslash infty$ and an Archimedean valuation as $v\; |\; \backslash infty.$ Then assume all valuations to be non-trivial.

There is a one-to-one identification of valuations and absolute values. Fix a constant $C>1,$ the valuation $v$ is assigned the absolute value $|\backslash cdot|\_v,$ defined as:

:$\backslash forall\; x\; \backslash in\; K:\; \backslash quad\; |x|\_v\; :=$

\begin{cases}

C^{-v(x)} & x \neq 0 \\

0 & x=0

\end{cases}

Conversely, the absolute value $|\backslash cdot|$ is assigned the valuation $v\_$

:$\backslash forall\; x\; \backslash in\; K^\backslash times:\; \backslash quad\; v\_$

A place of $K$ is a representative of an equivalence class of valuations (or absolute values) of $K.$ Places corresponding to non-Archimedean valuations are called finite, whereas places corresponding to Archimedean valuations are called infinite. Infinite places of a global field form a finite set, which is denoted by $P\_\{\backslash infty\}.$

Define and let $\backslash widehat\{O\}^\{\backslash times\}$ be its group of units. Then

Let $L/K$ be a finite extension of the global field $K.$ Let $w$ be a place of $L$ and $v$ a place of $K.$ If the absolute value $|\backslash cdot|\_w$ restricted to $K$ is in the equivalence class of $v$, then $w$ lies above $v,$ which is denoted by $w\; |\; v,$ and defined as:

:$\backslash begin\{align\}$

L_v&:=\prod_{w | v} L_w,\\

\widetilde{O_v} &:=\prod_{w | v}O_w.

\end{align}

(Note that both products are finite.)

If $w|v$, $K\_v$ can be embedded in $L\_w.$ Therefore, $K\_v$ is embedded diagonally in $L\_v.$ With this embedding $L\_v$ is a commutative algebra over $K\_v$ with degree

:$\backslash sum\_\{w|v\}[L\_w:K\_v]=[L:K].$

The set of finite adeles of a global field $K,$ denoted $\backslash mathbb\{A\}\_\{K,\backslash text\{fin\}\},$ is defined as the restricted product of $K\_v$ with respect to the $O\_v:$

:

It is equipped with the restricted product topology, the topology generated by restricted open rectangles, which have the following form:

:

where $E$ is a finite set of (finite) places and $U\_v\; \backslash subset\; K\_v$ are open. With component-wise addition and multiplication $\backslash mathbb\{A\}\_\{K,\backslash text\{fin\}\}$ is also a ring.

The adele ring of a global field $K$ is defined as the product of $\backslash mathbb\{A\}\_\{K,\backslash text\{fin\}\}$ with the product of the completions of $K$ at its infinite places. The number of infinite places is finite and the completions are either $\backslash R$ or $\backslash C.$ In short:

:

With addition and multiplication defined as component-wise the adele ring is a ring. The elements of the adele ring are called adeles of $K.$ In the following, it is written as

:$\backslash mathbb\{A\}\_K=\; \{\backslash prod\_v\}^\text{'}\; K\_v,$

although this is generally not a restricted product.

Remark. Global function fields do not have any infinite places and therefore the finite adele ring equals the adele ring.

:Lemma. There is a natural embedding of $K$ into $\backslash mathbb\{A\}\_K$ given by the diagonal map: $a\; \backslash mapsto\; (a,a,\backslash ldots).$

Proof. If $a\; \backslash in\; K,$ then $a\; \backslash in\; O\_v^\{\backslash times\}$ for almost all $v.$ This shows the map is well-defined. It is also injective because the embedding of $K$ in $K\_v$ is injective for all $v.$

Remark. By identifying $K$ with its image under the diagonal map it is regarded as a subring of $\backslash mathbb\{A\}\_K.$ The elements of $K$ are called the principal adeles of $\backslash mathbb\{A\}\_K.$

Definition. Let $S$ be a set of places of $K.$ Define the set of the $S$-adeles of $K$ as

: $\backslash mathbb\{A\}\_\{K,S\}\; :=\; \{\backslash prod\_\{v\; \backslash in\; S\}\}^\text{'}\; K\_v.$

Furthermore, if

:$\backslash mathbb\{A\}\_K^S\; :=\; \{\backslash prod\_\{v\; \backslash notin\; S\}\}^\text{'}\; K\_v$

the result is: $\backslash mathbb\{A\}\_K=\backslash mathbb\{A\}\_\{K,S\}\; \backslash times\; \backslash mathbb\{A\}\_K^S.$

By Ostrowski's theorem the places of $\backslash Q$ are $\backslash \{p\; \backslash in\; \backslash N\; :p\; \backslash text\{\; prime\}\backslash \}\; \backslash cup\; \backslash \{\backslash infty\backslash \},$ it is possible to identify a prime $p$ with the equivalence class of the $p$-adic absolute value and $\backslash infty$ with the equivalence class of the absolute value $|\backslash cdot|\_\backslash infty$ defined as:

:$\backslash forall\; x\; \backslash in\; \backslash Q:\; \backslash quad\; |x|\_\backslash infty:=$

\begin{cases}

x & x \geq 0 \\

-x & x < 0

\end{cases}

The completion of $\backslash Q$ with respect to the place $p$ is $\backslash Q\_p$ with valuation ring $\backslash Z\_p.$ For the place $\backslash infty$ the completion is $\backslash R.$ Thus:

:$\backslash begin\{align\}$

\mathbb{A}_{\Q,\text{fin}} &= {\prod_{p < \infty}}^' \Q_p \\

\mathbb{A}_{\Q} &= \left( {\prod_{p < \infty}}^' \Q_p \right) \times \R

\end{align}

Or for short

: $\backslash mathbb\{A\}\_\{\backslash Q\}\; =\; \{\backslash prod\_\{p\; \backslash leq\; \backslash infty\}\}^\text{'}\; \backslash Q\_p,\backslash qquad\; \backslash Q\_\backslash infty:=\backslash R.$

the difference between restricted and unrestricted product topology can be illustrated using a sequence in $\backslash mathbb\{A\}\_\backslash Q$:

:Lemma. Consider the following sequence in $\backslash mathbb\{A\}\_\backslash Q$:

::$\backslash begin\{align\}$

x_1&=\left(\frac 1 2 ,1,1,\ldots\right)\\

x_2&=\left(1,\frac 1 3 ,1,\ldots\right)\\

x_3&=\left(1,1,\frac 1 5 ,1,\ldots\right)\\

x_4&=\left(1,1,1,\frac 1 7 ,1,\ldots\right)\\

& \vdots

\end{align}

:In the product topology this converges to $(1,1,\backslash ldots)$, but it does not converge at all in the restricted product topology.

Proof. In product topology convergence corresponds to the convergence in each coordinate, which is trivial because the sequences become stationary. The sequence doesn't converge in restricted product topology. For each adele $a=(a\_p)\_p\; \backslash in\; \backslash mathbb\{A\}\_\{\backslash Q\}$ and for each restricted open rectangle $\backslash textstyle\; U=\backslash prod\_\{p\; \backslash in\; E\}U\_p\; \backslash times\; \backslash prod\_\{p\; \backslash notin\; E\}\backslash Z\_p,$ it has: $\backslash tfrac\{1\}\{p\}-a\_p\; \backslash notin\; \backslash Z\_p$ for $a\_p\; \backslash in\; \backslash Z\_p$ and therefore $\backslash tfrac\{1\}\{p\}-a\_p\; \backslash notin\; \backslash Z\_p$ for all $p\; \backslash notin\; F.$ As a result $x\_n-a\; \backslash notin\; U$ for almost all $n\; \backslash in\; \backslash N.$ In this consideration, $E$ and $F$ are finite subsets of the set of all places.

Definition (profinite integers). The profinite integers are defined as the profinite completion of the rings $\backslash Z\; /n\backslash Z$ with the partial order $n\; \backslash geq\; m\; \backslash Leftrightarrow\; m\; |\; n,$ i.e.,

:$\backslash widehat\{\backslash Z\}:=\backslash varprojlim\_n\; \backslash Z\; /n\backslash Z,$

:Lemma. $\backslash textstyle\; \backslash widehat\{\backslash Z\}\; \backslash cong\; \backslash prod\_p\; \backslash Z\_p.$

Proof. This follows from the Chinese Remainder Theorem.

:Lemma. $\backslash mathbb\{A\}\_\{\backslash Q,\; \backslash text\{fin\}\}=\; \backslash widehat\{\backslash Z\}\backslash otimes\_\{\backslash Z\}\; \backslash Q.$

Proof. Use the universal property of the tensor product. Define a $\backslash Z$-bilinear function

:$\backslash begin\{cases\}\; \backslash Psi:\; \backslash widehat\{\backslash Z\}\backslash times\; \backslash Q\; \backslash to\; \backslash mathbb\{A\}\_\{\backslash Q,\backslash text\{fin\}\}\; \backslash \backslash \; \backslash left\; ((a\_p)\_p,q\; \backslash right\; )\; \backslash mapsto\; (a\_pq)\_p\; \backslash end\{cases\}$

This is well-defined because for a given $q\; =\; \backslash tfrac\{m\}\{n\}\; \backslash in\; \backslash Q$ with $m,n$ co-prime there are only finitely many primes dividing $n.$ Let $M$ be another $\backslash Z$-module with a $\backslash Z$-bilinear map $\backslash Phi:\; \backslash widehat\{\backslash Z\}\; \backslash times\; \backslash Q\; \backslash to\; M.$ It must be the case that $\backslash Phi$ factors through $\backslash Psi$ uniquely, i.e., there exists a unique $\backslash Z$-linear map $\backslash tilde\{\backslash Phi\}:\; \backslash mathbb\{A\}\_\{\backslash Q,\backslash text\{fin\}\}\; \backslash to\; M$ such that $\backslash Phi\; =\; \backslash tilde\{\backslash Phi\}\; \backslash circ\; \backslash Psi.$ $\backslash tilde\{\backslash Phi\}$ can be defined as follows: for a given $(u\_p)\_p$ there exist $u\; \backslash in\; \backslash N$ and $(v\_p)\_p\; \backslash in\; \backslash widehat\{\backslash Z\}$ such that $u\_p=\backslash tfrac\{1\}\{u\}\backslash cdot\; v\_p$ for all $p.$ Define $\backslash tilde\{\backslash Phi\}((u\_p)\_p)\; :=\; \backslash Phi((v\_p)\_p,\; \backslash tfrac\{1\}\{u\}).$ One can show $\backslash tilde\{\backslash Phi\}$ is well-defined, $\backslash Z$-linear, satisfies $\backslash Phi\; =\; \backslash tilde\{\backslash Phi\}\; \backslash circ\; \backslash Psi$ and is unique with these properties.

:Corollary. Define $\backslash mathbb\{A\}\_\backslash Z\; :=\; \backslash widehat\{\backslash Z\}\; \backslash times\; \backslash R.$ This results in an algebraic isomorphism $\backslash mathbb\{A\}\_\{\backslash Q\}\; \backslash cong\; \backslash mathbb\{A\}\_\{\backslash Z\}\backslash otimes\_\{\backslash Z\}\; \backslash Q.$

Proof. $\backslash mathbb\{A\}\_\backslash Z\; \backslash otimes\_\backslash Z\; \backslash Q\; =\; \backslash left\; (\backslash widehat\{\backslash Z\}\backslash times\; \backslash R\; \backslash right\; )\backslash otimes\_\backslash Z\; \backslash Q\; \backslash cong\; \backslash left\; (\backslash widehat\{\backslash Z\}\; \backslash otimes\_\backslash Z\; \backslash Q\; \backslash right\; )\backslash times\; (\backslash R\; \backslash otimes\_\backslash Z\; \backslash Q)\; \backslash cong\; \backslash left\; (\backslash widehat\{\backslash Z\}\backslash otimes\_\{\backslash Z\}\; \backslash Q\; \backslash right\; )\; \backslash times\; \backslash R\; =\; \backslash mathbb\{A\}\_\{\backslash Q,\backslash text\{fin\}\}\; \backslash times\; \backslash R\; =\; \backslash mathbb\{A\}\_\{\backslash Q\}.$

:Lemma. For a number field $K,\; \backslash mathbb\{A\}\_K=\backslash mathbb\{A\}\_\{\backslash Q\}\backslash otimes\_\{\backslash Q\}\; K.$

Remark. Using $\backslash mathbb\{A\}\_\{\backslash Q\}\backslash otimes\_\{\backslash Q\}\; K\; \backslash cong\; \backslash mathbb\{A\}\_\{\backslash Q\}\; \backslash oplus\; \backslash dots\; \backslash oplus\; \backslash mathbb\{A\}\_\{\backslash Q\},$ where there are $[K:\backslash Q]$ summands, give the right side receives the product topology and transport this topology via the isomorphism onto $\backslash mathbb\{A\}\_\{\backslash Q\}\backslash otimes\_\{\backslash Q\}\; K.$

If $L/K$ be a finite extension, then $L$ is a global field. Thus $\backslash mathbb\{A\}\_L$ is defined, and $\backslash textstyle\; \backslash mathbb\{A\}\_L=\; \{\backslash prod\_v\}^\text{'}\; L\_v.$ $\backslash mathbb\{A\}\_K$ can be identified with a subgroup of $\backslash mathbb\{A\}\_L.$ Map $a=(a\_v)\_v\; \backslash in\; \backslash mathbb\{A\}\_K$ to $a\text{'}=(a\text{'}\_w)\_w\; \backslash in\; \backslash mathbb\{A\}\_L$ where $a\text{'}\_w=a\_v\; \backslash in\; K\_v\; \backslash subset\; L\_w$ for $w|v.$ Then $a=(a\_w)\_w\; \backslash in\; \backslash mathbb\{A\}\_L$ is in the subgroup $\backslash mathbb\{A\}\_K,$ if $a\_w\; \backslash in\; K\_v$ for $w\; |\; v$ and $a\_w=a\_\{w\text{'}\}$ for all $w,\; w\text{'}$ lying above the same place $v$ of $K.$

:Lemma. If $L/K$ is a finite extension, then $\backslash mathbb\{A\}\_L\backslash cong\backslash mathbb\{A\}\_K\; \backslash otimes\_K\; L$ both algebraically and topologically.

With the help of this isomorphism, the inclusion $\backslash mathbb\{A\}\_K\; \backslash subset\; \backslash mathbb\{A\}\_L$ is given by

:$\backslash begin\{cases\}$

\mathbb{A}_K \to \mathbb{A}_L\\

\alpha \mapsto \alpha \otimes_K 1

\end{cases}

Furthermore, the principal adeles in $\backslash mathbb\{A\}\_K$ can be identified with a subgroup of principal adeles in $\backslash mathbb\{A\}\_L$ via the map

:$\backslash begin\{cases\}$

K \to (K \otimes_K L) \cong L\\

\alpha \mapsto 1 \otimes_K \alpha

\end{cases}

Proof.This proof can be found in {{harvnb|Cassels|Fröhlich|1967|p=64.}} Let $\backslash omega\_1,\backslash ldots,\; \backslash omega\_n$ be a basis of $L$ over $K.$ Then for almost all $v,$

:$\backslash widetilde\{O\_v\}\; \backslash cong\; O\_v\backslash omega\_1\; \backslash oplus\; \backslash cdots\; \backslash oplus\; O\_v\; \backslash omega\_n.$

Furthermore, there are the following isomorphisms:

: $K\_v\backslash omega\_1\; \backslash oplus\; \backslash cdots\; \backslash oplus\; K\_v\; \backslash omega\_n\; \backslash cong\; K\_v\; \backslash otimes\_K\; L\; \backslash cong\; L\_v=\backslash prod\backslash nolimits\_\{w\; |\; v\}\; L\_w$

For the second use the map:

:$\backslash begin\{cases\}\; K\_v\; \backslash otimes\_K\; L\; \backslash to\; L\_v\; \backslash \backslash \backslash alpha\_v\; \backslash otimes\; a\; \backslash mapsto\; (\backslash alpha\_v\; \backslash cdot\; (\backslash tau\_w(a)))\_w\; \backslash end\{cases\}$

in which $\backslash tau\_w\; :\; L\; \backslash to\; L\_w$ is the canonical embedding and $w\; |\; v.$ The restricted product is taken on both sides with respect to $\backslash widetilde\{O\_v\}:$

: $\backslash begin\{align\}$

\mathbb{A}_K \otimes_K L &= \left ( {\prod_v}^' K_v \right ) \otimes_K L\\

&\cong {\prod_v}^' (K_v\omega_1 \oplus \cdots \oplus K_v \omega_n)\\

&\cong {\prod_v}^' (K_v \otimes_K L)\\

&\cong {\prod_v}^' L_v \\

&=\mathbb{A}_L

\end{align}

:Corollary. As additive groups $\backslash mathbb\{A\}\_L\; \backslash cong\; \backslash mathbb\{A\}\_K\; \backslash oplus\; \backslash cdots\; \backslash oplus\; \backslash mathbb\{A\}\_K,$ where the right side has $[L:K]$ summands.

The set of principal adeles in $\backslash mathbb\{A\}\_L$ is identified with the set $K\; \backslash oplus\; \backslash cdots\; \backslash oplus\; K,$ where the left side has $[L:K]$ summands and $K$ is considered as a subset of $\backslash mathbb\{A\}\_K.$

:Lemma. Suppose $P\backslash supset\; P\_\{\backslash infty\}$ is a finite set of places of $K$ and define

::$\backslash mathbb\{A\}\_K(P):=\backslash prod\_\{v\; \backslash in\; P\}\; K\_v\; \backslash times\; \backslash prod\_\{v\; \backslash notin\; P\}\; O\_v.$

:Equip $\backslash mathbb\{A\}\_K(P)$ with the product topology and define addition and multiplication component-wise. Then $\backslash mathbb\{A\}\_K(P)$ is a locally compact topological ring.

Remark. If $P\text{'}$ is another finite set of places of $K$ containing $P$ then $\backslash mathbb\{A\}\_K(P)$ is an open subring of $\backslash mathbb\{A\}\_K(P\text{'}).$

Now, an alternative characterisation of the adele ring can be presented. The adele ring is the union of all sets $\backslash mathbb\{A\}\_K(P)$:

:

Equivalently $\backslash mathbb\{A\}\_K$ is the set of all $x=(x\_v)\_v$ so that $|x\_v|\_v\; \backslash leq\; 1$ for almost all The topology of $\backslash mathbb\{A\}\_K$ is induced by the requirement that all $\backslash mathbb\{A\}\_K(P)$ be open subrings of $\backslash mathbb\{A\}\_K.$ Thus, $\backslash mathbb\{A\}\_K$ is a locally compact topological ring.

Fix a place $v$ of $K.$ Let $P$ be a finite set of places of $K,$ containing $v$ and $P\_\backslash infty.$ Define

:$\backslash mathbb\{A\}\_K\text{'}(P,v)\; :=\; \backslash prod\_\{w\; \backslash in\; P\; \backslash setminus\; \backslash \{v\backslash \}\}\; K\_w\; \backslash times\; \backslash prod\_\{w\; \backslash notin\; P\}\; O\_w.$

Then:

:$\backslash mathbb\{A\}\_K(P)\; \backslash cong\; K\_v\; \backslash times\; \backslash mathbb\{A\}\_K\text{'}(P,v).$

Furthermore, define

:$\backslash mathbb\{A\}\_K\text{'}(v):=\backslash bigcup\_\{P\; \backslash supset\; P\_\{\backslash infty\}\; \backslash cup\; \backslash \{v\backslash \}\}\; \backslash mathbb\{A\}\_K\text{'}(P,v),$

where $P$ runs through all finite sets containing $P\_\{\backslash infty\}\; \backslash cup\; \backslash \{v\backslash \}.$ Then:

:$\backslash mathbb\{A\}\_K\; \backslash cong\; K\_v\; \backslash times\; \backslash mathbb\{A\}\_K\text{'}(v),$

via the map $(a\_w)\_w\; \backslash mapsto\; (a\_v,\; (a\_w)\_\{w\; \backslash neq\; v\}).$ The entire procedure above holds with a finite subset $\backslash widetilde\{P\}$ instead of $\backslash \{v\backslash \}.$

By construction of $\backslash mathbb\{A\}\_K\text{'}(v),$ there is a natural embedding: $K\_v\; \backslash hookrightarrow\; \backslash mathbb\{A\}\_K.$ Furthermore, there exists a natural projection $\backslash mathbb\{A\}\_K\; \backslash twoheadrightarrow\; K\_v.$

Let $E$ be a finite dimensional vector-space over $K$ and $\backslash \{\backslash omega\_1,\backslash ldots,\backslash omega\_n\backslash \}$ a basis for $E$ over $K.$ For each place $v$ of $K$:

:$\backslash begin\{align\}$

E_v &:=E \otimes_K K_v \cong K_v\omega_1 \oplus \cdots \oplus K_v\omega_n \\

\widetilde{O_v} &:=O_v\omega_1 \oplus \cdots \oplus O_v\omega_n

\end{align}

The adele ring of $E$ is defined as

:$\backslash mathbb\{A\}\_E:=\; \{\backslash prod\_v\}^\text{'}\; E\_v.$

This definition is based on the alternative description of the adele ring as a tensor product equipped with the same topology that was defined when giving an alternate definition of adele ring for number fields. Next, $\backslash mathbb\{A\}\_E$ is equipped with the restricted product topology. Then $\backslash mathbb\{A\}\_E\; =\; E\; \backslash otimes\_K\; \backslash mathbb\{A\}\_K$ and $E$ is embedded in $\backslash mathbb\{A\}\_E$ naturally via the map $e\; \backslash mapsto\; e\; \backslash otimes\; 1.$

An alternative definition of the topology on $\backslash mathbb\{A\}\_E$ can be provided. Consider all linear maps: $E\; \backslash to\; K.$ Using the natural embeddings $E\; \backslash to\; \backslash mathbb\{A\}\_E$ and $K\; \backslash to\; \backslash mathbb\{A\}\_K,$ extend these linear maps to: $\backslash mathbb\{A\}\_E\; \backslash to\; \backslash mathbb\{A\}\_K.$ The topology on $\backslash mathbb\{A\}\_E$ is the coarsest topology for which all these extensions are continuous.

The topology can be defined in a different way. Fixing a basis for $E$ over $K$ results in an isomorphism $E\; \backslash cong\; K^n.$ Therefore fixing a basis induces an isomorphism $(\backslash mathbb\{A\}\_K)^n\; \backslash cong\; \backslash mathbb\{A\}\_E.$ The left-hand side is supplied with the product topology and transport this topology with the isomorphism onto the right-hand side. The topology doesn't depend on the choice of the basis, because another basis defines a second isomorphism. By composing both isomorphisms, a linear homeomorphism which transfers the two topologies into each other is obtained. More formally

:$\backslash begin\{align\}$

\mathbb{A}_E &= E \otimes_K \mathbb{A}_K\\

&\cong (K \otimes_K \mathbb{A}_K) \oplus \cdots \oplus (K \otimes_K \mathbb{A}_K)\\

&\cong \mathbb{A}_K \oplus \cdots \oplus \mathbb{A}_K

\end{align}

where the sums have $n$ summands. In case of $E=L,$ the definition above is consistent with the results about the adele ring of a finite extension $L/K.$

The definitions are based on {{harvnb|Weil|1967|p=60.}}

Let $A$ be a finite-dimensional algebra over $K.$ In particular, $A$ is a finite-dimensional vector-space over $K.$ As a consequence, $\backslash mathbb\{A\}\_\{A\}$ is defined and $\backslash mathbb\{A\}\_A\; \backslash cong\; \backslash mathbb\{A\}\_K\; \backslash otimes\_K\; A.$ Since there is multiplication on $\backslash mathbb\{A\}\_K$ and $A,$ a multiplication on $\backslash mathbb\{A\}\_A$ can be defined via:

:$\backslash forall\; \backslash alpha,\; \backslash beta\; \backslash in\; \backslash mathbb\{A\}\_K\; \backslash text\{\; and\; \}\; \backslash forall\; a,b\; \backslash in\; A:\; \backslash qquad\; (\backslash alpha\; \backslash otimes\_K\; a)\; \backslash cdot\; (\backslash beta\; \backslash otimes\_K\; b):=(\backslash alpha\backslash beta)\backslash otimes\_K(ab).$

As a consequence, $\backslash mathbb\{A\}\_\{A\}$ is an algebra with a unit over $\backslash mathbb\{A\}\_K.$ Let $\backslash mathcal\{B\}$ be a finite subset of $A,$ containing a basis for $A$ over $K.$ For any finite place $v$ , $M\_v$ is defined as the $O\_v$-module generated by $\backslash mathcal\{B\}$ in $A\_v.$ For each finite set of places, $P\backslash supset\; P\_\{\backslash infty\},$ define

:$\backslash mathbb\{A\}\_\{A\}(P,\backslash alpha)\; =\backslash prod\_\{v\; \backslash in\; P\}\; A\_v\; \backslash times\; \backslash prod\_\{v\; \backslash notin\; P\}\; M\_v.$

One can show there is a finite set $P\_0,$ so that $\backslash mathbb\{A\}\_\{A\}(P,\backslash alpha)$ is an open subring of $\backslash mathbb\{A\}\_\{A\},$ if $P\; \backslash supset\; P\_0.$ Furthermore $\backslash mathbb\{A\}\_\{A\}$ is the union of all these subrings and for $A=K,$ the definition above is consistent with the definition of the adele ring.

Let $L/K$ be a finite extension. Since $\backslash mathbb\{A\}\_K=\backslash mathbb\{A\}\_K\; \backslash otimes\_K\; K$ and $\backslash mathbb\{A\}\_L=\backslash mathbb\{A\}\_K\; \backslash otimes\_K\; L$ from the Lemma above, $\backslash mathbb\{A\}\_K$ can be interpreted as a closed subring of $\backslash mathbb\{A\}\_L.$ For this embedding, write $\backslash operatorname\{con\}\_\{L/K\}$. Explicitly for all places $w$ of $L$ above $v$ and for any $\backslash alpha\; \backslash in\; \backslash mathbb\{A\}\_K,\; (\backslash operatorname\{con\}\_\{L/K\}(\backslash alpha))\_w=\backslash alpha\_v\; \backslash in\; K\_v.$

Let $M/L/K$ be a tower of global fields. Then:

:$\backslash operatorname\{con\}\_\{M/K\}(\backslash alpha)=\backslash operatorname\{con\}\_\{M/L\}(\backslash operatorname\{con\}\_\{L/K\}(\backslash alpha))\; \backslash qquad\; \backslash forall\; \backslash alpha\; \backslash in\; \backslash mathbb\{A\}\_K.$

Furthermore, restricted to the principal adeles $\backslash operatorname\{con\}$ is the natural injection $K\; \backslash to\; L.$

Let $\backslash \{\backslash omega\_1,\backslash ldots,\backslash omega\_n\backslash \}$ be a basis of the field extension $L/K.$ Then each $\backslash alpha\; \backslash in\; \backslash mathbb\{A\}\_L$ can be written as $\backslash textstyle\; \backslash sum\_\{j=1\}^n\; \backslash alpha\_j\; \backslash omega\_j,$ where $\backslash alpha\_j\; \backslash in\; \backslash mathbb\{A\}\_K$ are unique. The map $\backslash alpha\; \backslash mapsto\; \backslash alpha\_j$ is continuous. Define $\backslash alpha\_\{ij\}$ depending on $\backslash alpha$ via the equations:

:$\backslash begin\{align\}$

\alpha \omega_1 &=\sum_{j=1}^n \alpha_{1j} \omega_j \\

&\vdots \\

\alpha \omega_n &=\sum_{j=1}^n \alpha_{nj} \omega_j

\end{align}

Now, define the trace and norm of $\backslash alpha$ as:

:$\backslash begin\{align\}$

\operatorname{Tr}_{L/K}(\alpha) &:= \operatorname{Tr} ((\alpha_{ij})_{i,j})=\sum_{i=1}^n \alpha_{ii}\\

N_{L/K}(\alpha) &:= N ((\alpha_{ij})_{i,j})=\det((\alpha_{ij})_{i,j})

\end{align}

These are the trace and the determinant of the linear map

:$\backslash begin\{cases\}\; \backslash mathbb\{A\}\_L\; \backslash to\; \backslash mathbb\{A\}\_L\; \backslash \backslash \; x\; \backslash mapsto\; \backslash alpha\; x\backslash end\{cases\}$

They are continuous maps on the adele ring, and they fulfil the usual equations:

: $\backslash begin\{align\}$

\operatorname{Tr}_{L/K}(\alpha+\beta)&=\operatorname{Tr}_{L/K}(\alpha) + \operatorname{Tr}_{L/K}(\beta) && \forall \alpha, \beta \in \mathbb{A}_L\\

\operatorname{Tr}_{L/K}(\operatorname{con}(\alpha))&=n\alpha && \forall \alpha \in \mathbb{A}_K\\

N_{L/K}(\alpha \beta)&=N_{L/K}(\alpha) N_{L/K}(\beta) && \forall \alpha, \beta \in \mathbb{A}_L\\

N_{L/K}(\operatorname{con}(\alpha))&=\alpha^n && \forall \alpha \in \mathbb{A}_K

\end{align}

Furthermore, for $\backslash alpha\; \backslash in\; L,$$\backslash operatorname\{Tr\}\_\{L/K\}(\backslash alpha)$ and $N\_\{L/K\}(\backslash alpha)$ are identical to the trace and norm of the field extension $L/K.$ For a tower of fields $M/L/K,$ the result is:

: $\backslash begin\{align\}$

\operatorname{Tr}_{L/K}(\operatorname{Tr}_{M/L}(\alpha)) &= \operatorname{Tr}_{M/K}(\alpha) && \forall \alpha \in \mathbb{A}_M\\

N_{L/K} (N_{M/L}(\alpha))&=N_{M/K}(\alpha) && \forall \alpha \in \mathbb{A}_M

\end{align}

Moreover, it can be proven that:See {{harvnb|Weil|1967|p=64}} or {{harvnb|Cassels|Fröhlich|1967|p=74}}.

:$\backslash begin\{align\}$

\operatorname{Tr}_{L/K}(\alpha) &= \left (\sum_{w | v}\operatorname{Tr}_{L_w/K_v}(\alpha_w) \right )_v && \forall \alpha \in \mathbb{A}_L\\

N_{L/K}(\alpha) &= \left (\prod_{w | v}N_{L_w/K_v}(\alpha_w) \right )_v && \forall \alpha \in \mathbb{A}_L

\end{align}

:Theorem.For proof see {{harvnb|Deitmar|2010|p=124}}, theorem 5.2.1. For every set of places $S,\; \backslash mathbb\{A\}\_\{K,S\}$ is a locally compact topological ring.

Remark. The result above also holds for the adele ring of vector-spaces and algebras over $K.$

:Theorem.See {{harvnb|Cassels|Fröhlich|1967|p=64}}, Theorem, or {{harvnb|Weil|1967|p=64}}, Theorem 2. $K$ is discrete and cocompact in $\backslash mathbb\{A\}\_K.$ In particular, $K$ is closed in $\backslash mathbb\{A\}\_K.$

Proof. Prove the case $K=\backslash Q.$ To show $\backslash Q\backslash subset\; \backslash mathbb\{A\}\_\backslash Q$ is discrete it is sufficient to show the existence of a neighbourhood of $0$ which contains no other rational number. The general case follows via translation. Define

:

$U$ is an open neighbourhood of $0\; \backslash in\; \backslash mathbb\{A\}\_\backslash Q.$ It is claimed that $U\; \backslash cap\; \backslash Q\; =\; \backslash \{0\backslash \}.$ Let $\backslash beta\; \backslash in\; U\; \backslash cap\; \backslash Q,$ then $\backslash beta\; \backslash in\; \backslash Q$ and $|\backslash beta|\_p\; \backslash leq\; 1$ for all $p$ and therefore $\backslash beta\; \backslash in\; \backslash Z.$ Additionally, $\backslash beta\; \backslash in\; (-1,1)$ and therefore $\backslash beta=0.$ Next, to show compactness, define:

:

Each element in $\backslash mathbb\{A\}\_\backslash Q\; /\backslash Q$ has a representative in $W,$ that is for each $\backslash alpha\; \backslash in\; \backslash mathbb\{A\}\_\backslash Q,$ there exists $\backslash beta\; \backslash in\; \backslash Q$ such that $\backslash alpha\; -\; \backslash beta\; \backslash in\; W.$ Let $\backslash alpha=(\backslash alpha\_p)\_p\; \backslash in\; \backslash mathbb\{A\}\_\backslash Q,$ be arbitrary and $p$ be a prime for which $|\backslash alpha\_p|>1.$ Then there exists $r\_p=z\_p/p^\{x\_p\}$ with $z\_p\; \backslash in\; \backslash Z,\; x\_p\; \backslash in\; \backslash N$ and $|\backslash alpha\_p-r\_p|\backslash leq\; 1.$ Replace $\backslash alpha$ with $\backslash alpha-r\_p$ and let $q\; \backslash neq\; p$ be another prime. Then:

:$\backslash left\; |\backslash alpha\_q-r\_p\; \backslash right\; |\_q\; \backslash leq\; \backslash max\; \backslash left\; \backslash \{|a\_q|\_q,|r\_p|\_q\; \backslash right\; \backslash \}\; \backslash leq\; \backslash max\; \backslash left\; \backslash \{|a\_q|\_q,1\; \backslash right\; \backslash \}\; \backslash leq\; 1.$

Next, it can be claimed that:

:$|\backslash alpha\_q-r\_p|\_q\; \backslash leq\; 1\; \backslash Longleftrightarrow\; |\backslash alpha\_q|\_q\; \backslash leq\; 1.$

The reverse implication is trivially true. The implication is true, because the two terms of the strong triangle inequality are equal if the absolute values of both integers are different. As a consequence, the (finite) set of primes for which the components of $\backslash alpha$ are not in $\backslash Z\_p$ is reduced by 1. With iteration, it can be deduced that there exists $r\backslash in\; \backslash Q$ such that $\backslash alpha-r\; \backslash in\; \backslash widehat\{\backslash Z\}\; \backslash times\; \backslash R.$ Now select $s\; \backslash in\; \backslash Z$ such that $\backslash alpha\_\backslash infty-r-s\; \backslash in\; \backslash left\; [-\backslash tfrac\{1\}\{2\},\; \backslash tfrac\{1\}\{2\}\; \backslash right\; ].$ Then $\backslash alpha-(r+s)\; \backslash in\; W.$ The continuous projection $\backslash pi:W\; \backslash to\backslash mathbb\{A\}\_\backslash Q\; /\backslash Q$ is surjective, therefore $\backslash mathbb\{A\}\_\backslash Q\; /\backslash Q,$ as the continuous image of a compact set, is compact.

:Corollary. Let $E$ be a finite-dimensional vector-space over $K.$ Then $E$ is discrete and cocompact in $\backslash mathbb\{A\}\_E.$

:Theorem. The following are assumed:

:*$\backslash mathbb\{A\}\_\{\backslash Q\}=\; \backslash Q\; +\backslash mathbb\{A\}\_\{\backslash Z\}.$

:*$\backslash Z\; =\backslash Q\; \backslash cap\; \backslash mathbb\{A\}\_\{\backslash Z\}.$

:*$\backslash mathbb\{A\}\_\{\backslash Q\}/\backslash Z$ is a divisible group.The next statement can be found in {{harvnb|Neukirch|2007|p=383}}.

:*$\backslash Q\; \backslash subset\; \backslash mathbb\{A\}\_\{\backslash Q,\backslash text\{fin\}\}$ is dense.

Proof. The first two equations can be proved in an elementary way.

By definition $\backslash mathbb\{A\}\_\{\backslash Q\}/\backslash Z$ is divisible if for any $n\; \backslash in\; \backslash N$ and $y\; \backslash in\; \backslash mathbb\{A\}\_\{\backslash Q\}/\backslash Z$ the equation $nx=y$ has a solution $x\; \backslash in\; \backslash mathbb\{A\}\_\{\backslash Q\}/\backslash Z.$ It is sufficient to show $\backslash mathbb\{A\}\_\{\backslash Q\}$ is divisible but this is true since $\backslash mathbb\{A\}\_\{\backslash Q\}$ is a field with positive characteristic in each coordinate.

For the last statement note that $\backslash mathbb\{A\}\_\{\backslash Q,\backslash text\{fin\}\}=\backslash Q\; \backslash widehat\{\backslash Z\},$ because the finite number of denominators in the coordinates of the elements of $\backslash mathbb\{A\}\_\{\backslash Q,\backslash text\{fin\}\}$ can be reached through an element $q\; \backslash in\; \backslash Q.$ As a consequence, it is sufficient to show $\backslash Z\; \backslash subset\; \backslash widehat\{\backslash Z\}$ is dense, that is each open subset $V\; \backslash subset\; \backslash widehat\{\backslash Z\}$ contains an element of $\backslash Z.$ Without loss of generality, it can be assumed that

:$V=\backslash prod\_\{p\; \backslash in\; E\}\; \backslash left(a\_p+p^\{l\_p\}\backslash Z\_p\; \backslash right\; )\; \backslash times\; \backslash prod\_\{p\; \backslash notin\; E\}\backslash Z\_p,$

because $(p^m\backslash Z\_p)\_\{m\; \backslash in\; \backslash N\}$ is a neighbourhood system of $0$ in $\backslash Z\_p.$ By Chinese Remainder Theorem there exists $l\; \backslash in\; \backslash Z$ such that $l\; \backslash equiv\; a\_p\; \backslash bmod\; p^\{l\_p\}.$ Since powers of distinct primes are coprime, $l\; \backslash in\; V$ follows.

Remark. $\backslash mathbb\{A\}\_\{\backslash Q\}/\backslash Z$ is not uniquely divisible. Let $y=(0,0,\backslash ldots)+\backslash Z\; \backslash in\; \backslash mathbb\{A\}\_\{\backslash Q\}/\backslash Z$ and $n\; \backslash geq\; 2$ be given. Then

:$\backslash begin\{align\}$

x_1 &=(0,0,\ldots)+\Z \\

x_2 &= \left (\tfrac{1}{n}, \tfrac{1}{n}, \ldots \right )+\Z

\end{align}

both satisfy the equation $nx=y$ and clearly $x\_1\; \backslash neq\; x\_2$ ($x\_2$ is well-defined, because only finitely many primes divide $n$). In this case, being uniquely divisible is equivalent to being torsion-free, which is not true for $\backslash mathbb\{A\}\_\{\backslash Q\}/\backslash Z$ since $nx\_2\; =\; 0,$ but $x\_2\; \backslash neq\; 0$ and $n\; \backslash neq\; 0.$

Remark. The fourth statement is a special case of the strong approximation theorem.

Definition. A function $f:\; \backslash mathbb\{A\}\_K\; \backslash to\; \backslash C$ is called simple if $\backslash textstyle\; f=\backslash prod\_v\; f\_v,$ where $f\_v:K\_v\; \backslash to\; \backslash C$ are measurable and $f\_v=\; \backslash mathbf\{1\}\_\{O\_v\}$ for almost all $v.$

:Theorem.See {{harvnb|Deitmar|2010|p=126}}, Theorem 5.2.2 for the rational case. Since $\backslash mathbb\{A\}\_K$ is a locally compact group with addition, there is an additive Haar measure $dx$ on $\backslash mathbb\{A\}\_K.$ This measure can be normalised such that every integrable simple function $\backslash textstyle\; f=\backslash prod\_v\; f\_v$ satisfies:

::$\backslash int\_\{\backslash mathbb\{A\}\_K\}\; f\; \backslash ,\; dx\; =\; \backslash prod\_v\; \backslash int\_\{K\_v\}\; f\_v\; \backslash ,\; dx\_v,$

:where for is the measure on $K\_v$ such that $O\_v$ has unit measure and $dx\_\{\backslash infty\}$ is the Lebesgue measure. The product is finite, i.e., almost all factors are equal to one.

Definition. Define the idele group of $K$ as the group of units of the adele ring of $K,$ that is $I\_K\; :=\; \backslash mathbb\{A\}\_K^\{\backslash times\}.$ The elements of the idele group are called the ideles of $K.$

Remark. $I\_K$ is equipped with a topology so that it becomes a topological group. The subset topology inherited from $\backslash mathbb\{A\}\_K$ is not a suitable candidate since the group of units of a topological ring equipped with subset topology may not be a topological group. For example, the inverse map in $\backslash mathbb\{A\}\_\{\backslash Q\}$ is not continuous. The sequence

:$\backslash begin\{align\}$

x_1&=(2,1,\ldots)\\

x_2&=(1,3,1,\ldots)\\

x_3&=(1,1,5,1,\ldots)\\

&\vdots

\end{align}

converges to $1\; \backslash in\; \backslash mathbb\{A\}\_\{\backslash Q\}.$ To see this let $U$ be neighbourhood of $0,$ without loss of generality it can be assumed:

:$U=\backslash prod\_\{p\; \backslash leq\; N\}\; U\_p\; \backslash times\; \backslash prod\_\{p\; >\; N\}\backslash Z\_p$

Since $(x\_n)\_p-1\; \backslash in\; \backslash Z\_p$ for all $p,$ $x\_n-1\; \backslash in\; U$ for $n$ large enough. However, as was seen above the inverse of this sequence does not converge in $\backslash mathbb\{A\}\_\{\backslash Q\}.$

:Lemma. Let $R$ be a topological ring. Define:

::$\backslash begin\{cases\}$

\iota: R^{\times} \to R \times R\\

x \mapsto (x,x^{-1})

\end{cases}

:Equipped with the topology induced from the product on topology on $R\; \backslash times\; R$ and $\backslash iota,\; R^\{\backslash times\}$ is a topological group and the inclusion map $R^\{\backslash times\}\; \backslash subset\; R$ is continuous. It is the coarsest topology, emerging from the topology on $R,$ that makes $R^\backslash times$ a topological group.

Proof. Since $R$ is a topological ring, it is sufficient to show that the inverse map is continuous. Let $U\backslash subset\; R^\backslash times$ be open, then $U\; \backslash times\; U^\{-1\}\; \backslash subset\; R\; \backslash times\; R$ is open. It is necessary to show $U^\{-1\}\; \backslash subset\; R^\backslash times$ is open or equivalently, that $U^\{-1\}\backslash times\; (U^\{-1\})^\{-1\}=U^\{-1\}\; \backslash times\; U\; \backslash subset\; R\; \backslash times\; R$ is open. But this is the condition above.

The idele group is equipped with the topology defined in the Lemma making it a topological group.

Definition. For $S$ a subset of places of $K$ set: $I\_\{K,S\}:=\backslash mathbb\{A\}\_\{K,S\}^\backslash times,\; I\_K^S:=(\backslash mathbb\{A\}\_\{K\}^S)^\{\backslash times\}.$

:Lemma. The following identities of topological groups hold:

::$\backslash begin\{align\}$

I_{K,S}&= {\prod_{v \in S}}^' K_v^{\times}\\

I_K^S&= {\prod_{v \notin S}}^' K_v^{\times}\\

I_K&= {\prod_v}^' K_v^{\times}

\end{align}

:where the restricted product has the restricted product topology, which is generated by restricted open rectangles of the form

::$\backslash prod\_\{v\; \backslash in\; E\}\; U\_v\; \backslash times\; \backslash prod\_\{v\; \backslash notin\; E\}\; O\_v^\{\backslash times\},$

:where $E$ is a finite subset of the set of all places and $U\_v\; \backslash subset\; K\_v^\{\backslash times\}$ are open sets.

Proof. Prove the identity for $I\_K$; the other two follow similarly. First show the two sets are equal:

:$\backslash begin\{align\}$

I_K &=\{x=(x_v)_v \in \mathbb{A}_K: \exists y=(y_v)_v \in \mathbb{A}_K: xy=1\}\\

&=\{x=(x_v)_v \in \mathbb{A}_K: \exists y=(y_v)_v \in \mathbb{A}_K: x_v \cdot y_v =1 \quad \forall v\} \\

&=\{x=(x_v)_v: x_v \in K_v^\times \forall v \text{ and } x_v \in O_v^\times \text{ for almost all } v\} \\

&= {\prod_v}^' K_v^\times

\end{align}

In going from line 2 to 3, $x$ as well as $x^\{-1\}=y$ have to be in $\backslash mathbb\{A\}\_K,$ meaning $x\_v\; \backslash in\; O\_v$ for almost all $v$ and $x\_v^\{-1\}\; \backslash in\; O\_v$ for almost all $v.$ Therefore, $x\_v\; \backslash in\; O\_v^\backslash times$ for almost all $v.$

Now, it is possible to show the topology on the left-hand side equals the topology on the right-hand side. Obviously, every open restricted rectangle is open in the topology of the idele group. On the other hand, for a given $U\; \backslash subset\; I\_K,$ which is open in the topology of the idele group, meaning $U\; \backslash times\; U^\{-1\}\; \backslash subset\; \backslash mathbb\{A\}\_K\; \backslash times\; \backslash mathbb\{A\}\_K$ is open, so for each $u\; \backslash in\; U$ there exists an open restricted rectangle, which is a subset of $U$ and contains $u.$ Therefore, $U$ is the union of all these restricted open rectangles and therefore is open in the restricted product topology.

:Lemma. For each set of places, $S,\; I\_\{K,S\}$ is a locally compact topological group.

Proof. The local compactness follows from the description of $I\_\{K,S\}$ as a restricted product. It being a topological group follows from the above discussion on the group of units of a topological ring.

A neighbourhood system of $1\; \backslash in\; \backslash mathbb\{A\}\_K(P\_\backslash infty)^\{\backslash times\}$ is a neighbourhood system of $1\; \backslash in\; I\_K.$ Alternatively, take all sets of the form:

:$\backslash prod\_v\; U\_v,$

where $U\_v$ is a neighbourhood of $1\; \backslash in\; K\_v^\backslash times$ and $U\_v=O\_v^\backslash times$ for almost all $v.$

Since the idele group is a locally compact, there exists a Haar measure $d^\backslash times\; x$ on it. This can be normalised, so that

:$\backslash int\_\{I\_\{K,\backslash text\{fin\}\}\}\; \backslash mathbf\{1\}\_\{\backslash widehat\{O\}\}\backslash ,\; d^\backslash times\; x\; =1.$

This is the normalisation used for the finite places. In this equation, $I\_\{K,\backslash text\{fin\}\}$ is the finite idele group, meaning the group of units of the finite adele ring. For the infinite places, use the multiplicative lebesgue measure $\backslash tfrac\{dx\}$

:Lemma. Let $L/K$ be a finite extension. Then:

::$I\_L=\; \{\backslash prod\_v\}^\text{'}\; L\_v^\backslash times.$

:where the restricted product is with respect to $\backslash widetilde\{O\_v\}^\{\backslash times\}.$

:Lemma. There is a canonical embedding of $I\_K$ in $I\_L.$

Proof. Map $a=(a\_v)\_v\; \backslash in\; I\_K$ to $a\text{'}=(a\text{'}\_w)\_w\; \backslash in\; I\_L$ with the property $a\text{'}\_w=a\_v\; \backslash in\; K\_v^\backslash times\; \backslash subset\; L\_w^\backslash times$ for $w|v.$ Therefore, $I\_K$ can be seen as a subgroup of $I\_L.$ An element $a=(a\_w)\_w\; \backslash in\; I\_L$ is in this subgroup if and only if his components satisfy the following properties: $a\_w\; \backslash in\; K\_v^\backslash times$ for $w\; |\; v$ and $a\_w=a\_\{w\text{'}\}$ for $w|v$ and $w\text{'}\; |\; v$ for the same place $v$ of $K.$

This section is based on {{harvnb|Weil|1967|p=71.}}

Let $A$ be a finite-dimensional algebra over $K.$ Since $\backslash mathbb\{A\}\_A^\{\backslash times\}$ is not a topological group with the subset-topology in general, equip $\backslash mathbb\{A\}\_\{A\}^\{\backslash times\}$ with the topology similar to $I\_K$ above and call $\backslash mathbb\{A\}\_A^\{\backslash times\}$ the idele group. The elements of the idele group are called idele of $A.$

:Proposition. Let $\backslash alpha$ be a finite subset of $A,$ containing a basis of $A$ over $K.$ For each finite place $v$ of $K,$ let $\backslash alpha\_v$ be the $O\_v$-module generated by $\backslash alpha$ in $A\_v.$ There exists a finite set of places $P\_0$ containing $P\_\{\backslash infty\}$ such that for all $v\; \backslash notin\; P\_0,$$\backslash alpha\_v$ is a compact subring of $A\_v.$ Furthermore, $\backslash alpha\_v$ contains $A\_v^\backslash times.$ For each $v,\; A\_v^\{\backslash times\}$ is an open subset of $A\_v$ and the map $x\; \backslash mapsto\; x^\{-1\}$ is continuous on $A\_v^\{\backslash times\}.$ As a consequence $x\; \backslash mapsto\; (x,x^\{-1\})$ maps $A\_v^\{\backslash times\}$ homeomorphically on its image in $A\_v\; \backslash times\; A\_v.$ For each $v\; \backslash notin\; P\_0,$ the $\backslash alpha\_v^\{\backslash times\}$ are the elements of $A\_v^\backslash times,$ mapping in $\backslash alpha\_v\; \backslash times\; \backslash alpha\_v$ with the function above. Therefore, $\backslash alpha\_v^\{\backslash times\}$ is an open and compact subgroup of $A\_v^\backslash times.$A proof of this statement can be found in {{harvnb|Weil|1967|p=71.}}

:Proposition. Let $P\; \backslash supset\; P\_\{\backslash infty\}$ be a finite set of places. Then

::$\backslash mathbb\{A\}\_\{A\}(P,\backslash alpha)^\{\backslash times\}:=\backslash prod\_\{v\; \backslash in\; P\}\; A\_v^\{\backslash times\}\; \backslash times\; \backslash prod\_\{v\; \backslash notin\; P\}\; \backslash alpha\_v^\{\backslash times\}$

:is an open subgroup of $\backslash mathbb\{A\}\_\{A\}^\{\backslash times\},$ where $\backslash mathbb\{A\}\_\{A\}^\{\backslash times\}$ is the union of all $\backslash mathbb\{A\}\_\{A\}(P,\backslash alpha)^\{\backslash times\}.$A proof of this statement can be found in {{harvnb|Weil|1967|p=72.}}

:Corollary. In the special case of $A=K$ for each finite set of places $P\; \backslash supset\; P\_\{\backslash infty\},$

::$\backslash mathbb\{A\}\_K(P)^\{\backslash times\}=\backslash prod\_\{v\; \backslash in\; P\}K\_v^\{\backslash times\}\; \backslash times\; \backslash prod\_\{v\; \backslash notin\; P\}O\_v^\{\backslash times\}$

:is an open subgroup of $\backslash mathbb\{A\}\_K^\{\backslash times\}=I\_K.$ Furthermore, $I\_K$ is the union of all $\backslash mathbb\{A\}\_K(P)^\{\backslash times\}.$

The trace and the norm should be transfer from the adele ring to the idele group. It turns out the trace can't be transferred so easily. However, it is possible to transfer the norm from the adele ring to the idele group. Let $\backslash alpha\; \backslash in\; I\_K.$ Then $\backslash operatorname\{con\}\_\{L/K\}(\backslash alpha)\; \backslash in\; I\_L$ and therefore, it can be said that in injective group homomorphism

:$\backslash operatorname\{con\}\_\{L/K\}:\; I\_K\; \backslash hookrightarrow\; I\_L.$

Since $\backslash alpha\; \backslash in\; I\_L,$ it is invertible, $N\_\{L/K\}(\backslash alpha)$ is invertible too, because $(N\_\{L/K\}(\backslash alpha))^\{-1\}=\; N\_\{L/K\}(\backslash alpha^\{-1\}).$ Therefore $N\_\{L/K\}(\backslash alpha)\; \backslash in\; I\_K.$ As a consequence, the restriction of the norm-function introduces a continuous function:

:$N\_\{L/K\}:\; I\_L\; \backslash to\; I\_K.$

:Lemma. There is natural embedding of $K^\{\backslash times\}$ into $I\_\{K,S\}$ given by the diagonal map: $a\; \backslash mapsto\; (a,a,a,\backslash ldots).$

Proof. Since $K^\{\backslash times\}$ is a subset of $K\_v^\{\backslash times\}$ for all $v,$ the embedding is well-defined and injective.

:Corollary. $A^\{\backslash times\}$ is a discrete subgroup of $\backslash mathbb\{A\}\_\{A\}^\{\backslash times\}.$

Defenition. In analogy to the ideal class group, the elements of $K^\{\backslash times\}$ in $I\_K$ are called principal ideles of $I\_K.$ The quotient group $C\_K\; :=\; I\_K/K^\{\backslash times\}$ is called idele class group of $K.$ This group is related to the ideal class group and is a central object in class field theory.

Remark. $K^\backslash times$ is closed in $I\_K,$ therefore $C\_K$ is a locally compact topological group and a Hausdorff space.

:Lemma.For a proof see {{harvnb|Neukirch|2007|p=388}}. Let $L/K$ be a finite extension. The embedding $I\_K\; \backslash to\; I\_L$ induces an injective map:

::$\backslash begin\{cases\}$

C_K \to C_L\\

\alpha K^\times \mapsto \alpha L^\times

\end{cases}

Definition. For $\backslash alpha=(\backslash alpha\_v)\_v\; \backslash in\; I\_K$ define: $\backslash textstyle\; |\backslash alpha|:=\backslash prod\_v\; |\backslash alpha\_v|\_v.$ Since $\backslash alpha$ is an idele this product is finite and therefore well-defined.

Remark. The definition can be extended to $\backslash mathbb\{A\}\_K$ by allowing infinite products. However, these infinite products vanish and so $|\backslash cdot|$ vanishes on $\backslash mathbb\{A\}\_K\; \backslash setminus\; I\_K.$ $|\backslash cdot|$ will be used to denote both the function on $I\_K$ and $\backslash mathbb\{A\}\_K.$

:Theorem. $|\backslash cdot|:I\_K\; \backslash to\; \backslash R\_+$ is a continuous group homomorphism.

Proof. Let $\backslash alpha,\; \backslash beta\; \backslash in\; I\_K.$

: $\backslash begin\{align\}$

|\alpha \cdot \beta|&=\prod_v |(\alpha \cdot \beta)_v|_v\\

&=\prod_v|\alpha_v \cdot \beta_v|_v\\

&=\prod_v(|\alpha_v|_v \cdot |\beta_v|_v)\\

&=\left(\prod_v |\alpha_v|_v\right) \cdot \left(\prod_v |\beta_v|_v\right)\\

&= |\alpha|\cdot |\beta|

\end{align}

where it is used that all products are finite. The map is continuous which can be seen using an argument dealing with sequences. This reduces the problem to whether $|\backslash cdot|$ is continuous on $K\_v.$ However, this is clear, because of the reverse triangle inequality.

Definition. The set of $1$-idele can be defined as:

:$I\_K^1:=\backslash \{x\; \backslash in\; I\_K:\; |x|=1\backslash \}=\backslash ker(|\backslash cdot|).$

$I\_K^1$ is a subgroup of $I\_K.$ Since $I\_K^1=|\backslash cdot|^\{-1\}(\backslash \{1\backslash \}),$ it is a closed subset of $\backslash mathbb\{A\}\_K.$ Finally the $\backslash mathbb\{A\}\_K$-topology on $I\_K^1$ equals the subset-topology of $I\_K$ on $I\_K^1.$This statement can be found in {{harvnb|Cassels|Fröhlich|1967|p=69}}.$\backslash mathbb\{A\}\_K^1$ is also used for the set of the $1$-idele but $I\_K^1$ is used in this example.

:Artin's Product Formula. $|k|=1$ for all $k\; \backslash in\; K^\backslash times.$

Proof.There are many proofs for this result. The one shown below is based on {{harvnb|Neukirch|2007|p=195.}} Proof of the formula for number fields, the case of global function fields can be proved similarly. Let $K$ be a number field and $a\; \backslash in\; K^\backslash times.$ It has to be shown that:

:$\backslash prod\_v|a|\_v=1.$

For finite place $v$ for which the corresponding prime ideal $\backslash mathfrak\{p\}\_v$ does not divide $(a)$, $v(a)=0$ and therefore $|a|\_v=1.$ This is valid for almost all $\backslash mathfrak\{p\}\_v.$ There is:

: $\backslash begin\{align\}$

\prod_v|a|_v&=\prod_{p \leq \infty} \prod_{v| p}|a|_v\\

&=\prod_{p \leq \infty} \prod_{v| p}|N_{K_v/ \Q_p}(a)|_p\\

&=\prod_{p \leq \infty} |N_{K /\Q}(a)|_p

\end{align}

In going from line 1 to line 2, the identity $|a|\_w=|N\_\{L\_w/\; K\_v\}(a)|\_v,$ was used where $v$ is a place of $K$ and $w$ is a place of $L,$ lying above $v.$ Going from line 2 to line 3, a property of the norm is used. The norm is in $\backslash Q$ so without loss of generality it can be assumed that $a\; \backslash in\; \backslash Q.$ Then $a$ possesses a unique integer factorisation:

:

where $v\_p\; \backslash in\; \backslash Z$ is $0$ for almost all $p.$ By Ostrowski's theorem all absolute values on $\backslash Q$ are equivalent to the real absolute value $|\backslash cdot|\_\{\backslash infty\}$ or a $p$-adic absolute value. Therefore:

:$\backslash begin\{align\}$

|a| &= \left(\prod_{p < \infty} |a|_p\right) \cdot |a|_{\infty}\\

&= \left(\prod_{p < \infty} p^{-v_p}\right) \cdot \left(\prod_{p < \infty}p^{v_p}\right) \\

&= 1

\end{align}

:Lemma.For a proof see {{harvnb|Cassels|Fröhlich|1967|p=66.}} There exists a constant $C,$ depending only on $K,$ such that for every $\backslash alpha=(\backslash alpha\_v)\_v\; \backslash in\; \backslash mathbb\{A\}\_K$ satisfying $\backslash textstyle\; \backslash prod\_v\; |\backslash alpha\_v|\_v\; >\; C,$ there exists $\backslash beta\; \backslash in\; K^\{\backslash times\}$ such that $|\backslash beta\_v|\_v\backslash leq\; |\backslash alpha\_v|\_v$ for all $v.$

:Corollary. Let $v\_0$ be a place of $K$ and let $\backslash delta\_v\; >\; 0$ be given for all $v\; \backslash neq\; v\_0$ with the property $\backslash delta\_v=1$ for almost all $v.$ Then there exists $\backslash beta\; \backslash in\; K^\{\backslash times\},$ so that $|\backslash beta|\; \backslash leq\; \backslash delta\_v$ for all $v\; \backslash neq\; v\_0.$

Proof. Let $C$ be the constant from the lemma. Let $\backslash pi\_v$ be a uniformising element of $O\_v.$ Define the adele $\backslash alpha=(\backslash alpha\_v)\_v$ via $\backslash alpha\_v:=\backslash pi\_v^\{k\_v\}$ with $k\_v\; \backslash in\; \backslash Z$ minimal, so that $|\backslash alpha\_v|\_v\; \backslash leq\; \backslash delta\_v$ for all $v\; \backslash neq\; v\_0.$ Then $k\_v=0$ for almost all $v.$ Define $\backslash alpha\_\{v\_0\}:=\backslash pi\_\{v\_0\}^\{k\_\{v\_0\}\}$ with $k\_\{v\_0\}\backslash in\; \backslash Z,$ so that $\backslash textstyle\; \backslash prod\_v\; |\backslash alpha\_v|\_v\; >\; C.$ This works, because $k\_v=0$ for almost all $v.$ By the Lemma there exists $\backslash beta\; \backslash in\; K^\backslash times,$ so that $|\backslash beta|\_v\; \backslash leq\; |\backslash alpha\_v|\_v\; \backslash leq\; \backslash delta\_v$ for all $v\; \backslash neq\; v\_0.$

:Theorem. $K^\backslash times$ is discrete and cocompact in $I\_K^1.$

Proof.This proof can be found in {{harvnb|Weil|1967|p=76}} or in {{harvnb|Cassels|Fröhlich|1967|p=70}}. Since $K^\backslash times$ is discrete in $I\_K$ it is also discrete in $I\_K^1.$ To prove the compactness of $I\_K^1/K^\backslash times$ let $C$ is the constant of the Lemma and suppose $\backslash alpha\; \backslash in\; \backslash mathbb\{A\}\_K$ satisfying $\backslash textstyle\; \backslash prod\_v\; |\backslash alpha\_v|\_v\; >\; C$ is given. Define:

:$W\_\backslash alpha:=\; \backslash left\; \backslash \{\backslash xi=(\backslash xi\_v)\_v\; \backslash in\; \backslash mathbb\{A\}\_K\; |\; |\backslash xi\_v|\_v\backslash leq\; |\backslash alpha\_v|\_v\; \backslash text\{\; for\; all\; \}\; v\backslash right\; \backslash \}.$

Clearly $W\_\backslash alpha$ is compact. It can be claimed that the natural projection $W\_\{\backslash alpha\}\; \backslash cap\; I\_K^1\; \backslash to\; I\_K^1/K^\backslash times$ is surjective. Let $\backslash beta=(\backslash beta\_v)\_v\; \backslash in\; I\_K^1$ be arbitrary, then:

:$|\backslash beta|\; =\; \backslash prod\_v\; |\backslash beta\_v|\_v=1,$

and therefore

:$\backslash prod\_v\; |\backslash beta\_v^\{-1\}|\_v=1.$

It follows that

:$\backslash prod\_v\; |\backslash beta\_v^\{-1\}\backslash alpha\_v|\_v=\backslash prod\_v\; |\backslash alpha\_v|\_v>C.$

By the Lemma there exists $\backslash eta\; \backslash in\; K^\backslash times$ such that $|\backslash eta|\_v\; \backslash leq\; |\backslash beta\_v^\{-1\}\backslash alpha\_v|\_v$ for all $v,$ and therefore $\backslash eta\backslash beta\; \backslash in\; W\_\{\backslash alpha\}$ proving the surjectivity of the natural projection. Since it is also continuous the compactness follows.

:Theorem.Part of Theorem 5.3.3 in {{harvnb|Deitmar|2010}}. There is a canonical isomorphism $I\_\{\backslash Q\}^1/\backslash Q^\backslash times\; \backslash cong\; \backslash widehat\{\backslash Z\}^\backslash times.$ Furthermore, $\backslash widehat\{\backslash Z\}^\backslash times\; \backslash times\; \backslash \{1\backslash \}\; \backslash subset\; I\_\{\backslash Q\}^1$ is a set of representatives for $I\_\{\backslash Q\}^1/\backslash Q^\backslash times$ and $\backslash widehat\{\backslash Z\}^\backslash times\; \backslash times\; (0,\; \backslash infty)\; \backslash subset\; I\_\{\backslash Q\}$ is a set of representatives for $I\_\{\backslash Q\}/\backslash Q^\backslash times.$

Proof. Consider the map

:$\backslash begin\{cases\}\; \backslash phi:\; \backslash widehat\{\backslash Z\}^\backslash times\; \backslash to\; I\_\{\backslash Q\}^1/\backslash Q^\backslash times\; \backslash \backslash \; (a\_p)\_p\; \backslash mapsto\; ((a\_p)\_p,1)\backslash Q^\backslash times\; \backslash end\{cases\}$

This map is well-defined, since $|a\_p|\_p=1$ for all $p$ and therefore Obviously $\backslash phi$ is a continuous group homomorphism. Now suppose $((a\_p)\_p,1)\backslash Q^\backslash times=((b\_p)\_p,1)\backslash Q^\backslash times.$ Then there exists $q\; \backslash in\; \backslash Q^\backslash times$ such that $((a\_p)\_p,1)q=((b\_p)\_p,1).$ By considering the infinite place it can be seen that $q=1$ proves injectivity. To show surjectivity let $((\backslash beta\_p)\_p,\; \backslash beta\_\backslash infty)\; \backslash Q^\backslash times\backslash in\; I\_\{\backslash Q\}^1/\backslash Q^\backslash times.$ The absolute value of this element is $1$ and therefore

:$|\backslash beta\_\backslash infty|\_\backslash infty=\backslash frac\{1\}\{\backslash prod\_p\; |\backslash beta\_p|\_p\}\; \backslash in\; \backslash Q.$

Hence $\backslash beta\_\backslash infty\; \backslash in\; \backslash Q$ and there is:

:$((\backslash beta\_p)\_p,\; \backslash beta\_\backslash infty)\; \backslash Q^\backslash times=\; \backslash left\; (\; \backslash left\; (\backslash frac\{\backslash beta\_p\}\{\backslash beta\_\backslash infty\}\; \backslash right\; )\_p,1\; \backslash right\; )\backslash Q^\backslash times.$

Since

:$\backslash forall\; p:\; \backslash qquad\; \backslash left\; |\; \backslash frac\{\backslash beta\_p\}\{\backslash beta\_\backslash infty\}\; \backslash right\; |\_p=1,$

It has been concluded that $\backslash phi$ is surjective.

:Theorem. The absolute value function induces the following isomorphisms of topological groups:

::$\backslash begin\{align\}$

I_{\Q} &\cong I_{\Q}^1 \times (0, \infty) \\

I_{\Q}^1 &\cong I_{\Q, \text{fin}} \times \{\pm 1\}.

\end{align}

Proof. The isomorphisms are given by:

:$\backslash begin\{cases\}\; \backslash psi:\; I\_\backslash Q\; \backslash to\; I\_\{\backslash Q\}^1\; \backslash times\; (0,\; \backslash infty)\; \backslash \backslash \; a=(a\_\backslash text\{fin\},\; a\_\{\backslash infty\})\; \backslash mapsto\; \backslash left\; (a\_\backslash text\{fin\},\backslash frac\{a\_\backslash infty\}$

:Theorem. Let $K$ be a number field with ring of integers $O,$ group of fractional ideals $J\_K,$ and ideal class group $\backslash operatorname\{Cl\}\_K\; =J\_K/K^\backslash times.$ Here's the following isomorphisms

::$\backslash begin\{align\}$

J_K &\cong I_{K,\text{fin}}/\widehat{O}^\times \\

\operatorname{Cl}_K &\cong C_{K,\text{fin}}/\widehat{O}^\times K^\times \\

\operatorname{Cl}_K &\cong C_K/\left (\widehat{O}^\times \times \prod_{v | \infty} K_v^\times \right ) K^\times

\end{align}

:where $C\_\{K,\backslash text\{fin\}\}\; :=I\_\{K,\backslash text\{fin\}\}/K^\backslash times$ has been defined.

Proof. Let $v$ be a finite place of $K$ and let $|\backslash cdot|\_v$ be a representative of the equivalence class $v.$ Define

:

Then $\backslash mathfrak\{p\}\_v$ is a prime ideal in $O.$ The map $v\; \backslash mapsto\; \backslash mathfrak\{p\}\_v$ is a bijection between finite places of $K$ and non-zero prime ideals of $O.$ The inverse is given as follows: a prime ideal $\backslash mathfrak\{p\}$ is mapped to the valuation $v\_\backslash mathfrak\{p\},$ given by

: $\backslash begin\{align\}$

v_\mathfrak{p}(x)&:= \max\{k \in \N _0: x \in \mathfrak{p}^k\} \quad \forall x \in O^\times \\

v_\mathfrak{p}\left( \frac{x}{y}\right) &:= v_\mathfrak{p}(x)-v_\mathfrak{p}(y) \quad \forall x,y \in O^\times

\end{align}

The following map is well-defined:

: $\backslash begin\{cases\}$

(\cdot): I_{K,\text{fin}} \to J_K\\

\alpha = (\alpha_v)_v \mapsto \prod_{v < \infty} \mathfrak{p}_v^{v(\alpha_v)},

\end{cases}

The map $(\backslash cdot)$ is obviously a surjective homomorphism and $\backslash ker((\backslash cdot))=\backslash widehat\{O\}^\backslash times.$ The first isomorphism follows from fundamental theorem on homomorphism. Now, both sides are divided by $K^\{\backslash times\}.$ This is possible, because

:$\backslash begin\{align\}$

(\alpha)&=((\alpha,\alpha,\dotsc)) \\

&=\prod_{v < \infty}\mathfrak{p}_v^{v(\alpha)}\\

&=(\alpha) && \text{ for all } \alpha \in K^\times.

\end{align}

Please, note the abuse of notation: On the left hand side in line 1 of this chain of equations, $(\backslash cdot)$ stands for the map defined above. Later, the embedding of $K^\{\backslash times\}$ into $I\_\{K,\backslash text\{fin\}\}$ is used. In line 2, the definition of the map is used. Finally, use

that $O$ is a Dedekind domain and therefore each ideal can be written as a product of prime ideals. In other words, the map $(\backslash cdot)$ is a $K^\backslash times$-equivariant group homomorphism. As a consequence, the map above induces a surjective homomorphism

: $\backslash begin\{cases\}$

\phi:C_{K,\text{fin}} \to \operatorname{Cl}_K\\

\alpha K^\times \mapsto (\alpha) K^\times

\end{cases}

To prove the second isomorphism, it has to be shown that $\backslash ker(\backslash phi)=\backslash widehat\{O\}^\{\backslash times\}K^\backslash times.$ Consider $\backslash xi=(\backslash xi\_v)\_v\; \backslash in\; \backslash widehat\{O\}^\{\backslash times\}.$ Then $\backslash textstyle\; \backslash phi(\backslash xi\; K^\backslash times)\; =\backslash prod\_v\; \backslash mathfrak\{p\}\_v^\{v(\backslash xi\_v)\}K^\backslash times=K^\backslash times,$ because $v(\backslash xi\_v)=0$ for all $v.$ On the other hand, consider $\backslash xi\; K^\backslash times\; \backslash in\; C\_\{K,\backslash text\{fin\}\}$ with $\backslash phi(\backslash xi\; K^\backslash times)=O\; K^\backslash times,$ which allows to write $\backslash textstyle\; \backslash prod\_v\backslash mathfrak\{p\}\_v^\{v(\backslash xi\_v)\}\; K^\backslash times=O\; K^\backslash times.$ As a consequence, there exists a representative, such that: $\backslash textstyle\; \backslash prod\_v\; \backslash mathfrak\{p\}\_v^\{v(\backslash xi\text{'}\_v)\}=O.$ Consequently, $\backslash xi\text{'}\; \backslash in\; \backslash widehat\{O\}^\backslash times$ and therefore $\backslash xi\; K^\backslash times=\backslash xi\text{'}\; K^\backslash times\; \backslash in\; \backslash widehat\{O\}^\backslash times\; K^\backslash times.$ The second isomorphism of the theorem has been proven.

For the last isomorphism note that $\backslash phi$ induces a surjective group homomorphism $\backslash widetilde\{\backslash phi\}:\; C\_K\; \backslash to\; \backslash operatorname\{Cl\}\_K$ with

:$\backslash ker(\backslash widetilde\{\backslash phi\})=\; \backslash left\; (\backslash widehat\{O\}^\backslash times\; \backslash times\; \backslash prod\_\{v\; |\; \backslash infty\}K\_v^\backslash times\; \backslash right\; )K^\backslash times.$

Remark. Consider $I\_\{K,\backslash text\{fin\}\}$ with the idele topology and equip $J\_K,$ with the discrete topology. Since $(\backslash \{\backslash mathfrak\{a\}\backslash \})^\{-1\}$ is open for each $\backslash mathfrak\{a\}\; \backslash in\; J\_K,\; (\backslash cdot)$ is continuous. It stands, that $(\backslash \{\backslash mathfrak\{a\}\backslash \})^\{-1\}\; =\; \backslash alpha\; \backslash widehat\{O\}^\backslash times$ is open, where $\backslash alpha=(\backslash alpha\_v)\_v\; \backslash in\; \backslash mathbb\{A\}\_\{K,\backslash text\{fin\}\},$ so that $\backslash textstyle\; \backslash mathfrak\{a\}=\backslash prod\_v\; \backslash mathfrak\{p\}\_v^\{v(\backslash alpha\_v)\}.$

:Theorem.

::$\backslash begin\{align\}$

I_K &\cong I_K^1 \times M: \quad \begin{cases} M \subset I_K \text{ discrete and } M \cong \Z & \operatorname{char}(K)>0 \\ M \subset I_K \text{ closed and } M \cong \R_+ & \operatorname{char}(K)=0 \end{cases} \\

C_K &\cong I_K^1/K^\times \times N: \quad \begin{cases} N = \Z & \operatorname{char}(K)>0 \\ N = \R_+ & \operatorname{char}(K)=0 \end{cases}

\end{align}

Proof. $\backslash operatorname\{char\}(K)\; =\; p>0.$ For each place $v$ of $K,\; \backslash operatorname\{char\}(K\_v)\; =\; p,$ so that for all $x\; \backslash in\; K\_v^\{\backslash times\},$ $|x|\_v$ belongs to the subgroup of $\backslash R\_+,$ generated by $p.$ Therefore for each $z\; \backslash in\; I\_K,$ $|z|$ is in the subgroup of $\backslash R\_+,$ generated by $p.$ Therefore the image of the homomorphism $z\; \backslash mapsto\; |z|$ is a discrete subgroup of $\backslash R\_+,$ generated by $p.$ Since this group is non-trivial, it is generated by $Q=p^m$ for some $m\; \backslash in\; \backslash N.$ Choose $z\_1\; \backslash in\; I\_K,$ so that $|z\_1|=Q,$ then $I\_K$ is the direct product of $I\_K^1$ and the subgroup generated by $z\_1.$ This subgroup is discrete and isomorphic to $\backslash Z.$

$\backslash operatorname\{char\}(K)\; =\; 0.$ For $\backslash lambda\; \backslash in\; \backslash R\_+$ define:

:

The map $\backslash lambda\; \backslash mapsto\; z(\backslash lambda)$ is an isomorphism of $\backslash R\_+$ in a closed subgroup $M$ of $I\_K$ and $I\_K\; \backslash cong\; M\; \backslash times\; I\_K^1.$ The isomorphism is given by multiplication:

: $\backslash begin\{cases\}$

\phi: M \times I_K^1 \to I_K,\\

((\alpha_v)_v, (\beta_v)_v) \mapsto (\alpha_v \beta_v)_v

\end{cases}

Obviously, $\backslash phi$ is a homomorphism. To show it is injective, let $(\backslash alpha\_v\; \backslash beta\_v)\_v=1.$ Since $\backslash alpha\_v=1$ for it stands that $\backslash beta\_v=1$ for Moreover, it exists a $\backslash lambda\; \backslash in\; \backslash R\_+,$ so that $\backslash alpha\_v=\backslash lambda$ for $v\; |\; \backslash infty.$ Therefore, $\backslash beta\_v=\backslash lambda^\{-1\}$ for $v\; |\; \backslash infty.$ Moreover $\backslash textstyle\; \backslash prod\_v\; |\backslash beta\_v|\_v\; =1,$ implies $\backslash lambda^n=1,$ where $n$ is the number of infinite places of $K.$ As a consequence $\backslash lambda=1$ and therefore $\backslash phi$ is injective. To show surjectivity, let $\backslash gamma=(\backslash gamma\_v)\_v\; \backslash in\; I\_K.$ It is defined that $\backslash lambda:=|\backslash gamma|^\{\backslash frac\{1\}\{n\}\}$ and furthermore, $\backslash alpha\_v=1$ for and $\backslash alpha\_v=\backslash lambda$ for $v\; |\; \backslash infty.$ Define $\backslash textstyle\; \backslash beta=\backslash frac\{\backslash gamma\}\{\backslash alpha\}.$ It stands, that $\backslash textstyle\; |\backslash beta|=\backslash frac$

The other equations follow similarly.

:Theorem.The general proof of this theorem for any global field is given in {{harvnb|Weil|1967|p=77.}} Let $K$ be a number field. There exists a finite set of places $S,$ such that:

::$I\_K=\; \backslash left\; (I\_\{K,S\}\; \backslash times\; \backslash prod\_\{v\; \backslash notin\; S\}\; O\_v^\backslash times\; \backslash right\; )\; K^\backslash times=\; \backslash left(\backslash prod\_\{v\; \backslash in\; S\}\; K\_v^\backslash times\; \backslash times\; \backslash prod\_\{v\; \backslash notin\; S\}\; O\_v^\backslash times\backslash right)\; K^\backslash times.$

Proof. The class number of a number field is finite so let $\backslash mathfrak\{a\}\_1,\; \backslash ldots,\; \backslash mathfrak\{a\}\_h$ be the ideals, representing the classes in $\backslash operatorname\{Cl\}\_K.$ These ideals are generated by a finite number of prime ideals $\backslash mathfrak\{p\}\_1,\; \backslash ldots,\; \backslash mathfrak\{p\}\_n.$ Let $S$ be a finite set of places containing $P\_\backslash infty$ and the finite places corresponding to $\backslash mathfrak\{p\}\_1,\; \backslash ldots,\; \backslash mathfrak\{p\}\_n.$ Consider the isomorphism:

:

induced by

:

At infinite places the statement is immediate, so the statement has been proved for finite places. The inclusion ″$\backslash supset$″ is obvious. Let $\backslash alpha\; \backslash in\; I\_\{K,\backslash text\{fin\}\}.$ The corresponding ideal belongs to a class $\backslash mathfrak\{a\}\_iK^\{\backslash times\},$ meaning $(\backslash alpha)=\backslash mathfrak\{a\}\_i(a)$ for a principal ideal $(a).$ The idele $\backslash alpha\text{'}=\backslash alpha\; a^\{-1\}$ maps to the ideal $\backslash mathfrak\{a\}\_i$ under the map $I\_\{K,\backslash text\{fin\}\}\; \backslash to\; J\_K.$ That means Since the prime ideals in $\backslash mathfrak\{a\}\_i$ are in $S,$ it follows $v(\backslash alpha\text{'}\_v)=0$ for all $v\; \backslash notin\; S,$ that means $\backslash alpha\text{'}\_v\; \backslash in\; O\_v^\backslash times$ for all $v\; \backslash notin\; S.$ It follows, that $\backslash alpha\text{'}=\backslash alpha\; a^\{-1\}\; \backslash in\; I\_\{K,S\},$ therefore $\backslash alpha\; \backslash in\; I\_\{K,S\}K^\backslash times.$

In the previous section the fact that the class number of a number field is finite had been used. Here this statement can be proved:

:Theorem (finiteness of the class number of a number field). Let $K$ be a number field. Then

Proof. The map

:

is surjective and therefore $\backslash operatorname\{Cl\}\_K$ is the continuous image of the compact set $I\_K^1/K^\{\backslash times\}.$ Thus, $\backslash operatorname\{Cl\}\_K$ is compact. In addition, it is discrete and so finite.

Remark. There is a similar result for the case of a global function field. In this case, the so-called divisor group is defined. It can be shown that the quotient of the set of all divisors of degree $0$ by the set of the principal divisors is a finite group.For more information, see {{harvnb|Cassels|Fröhlich|1967|p=71}}.

Let $P\; \backslash supset\; P\_\{\backslash infty\}$ be a finite set of places. Define

:$\backslash begin\{align\}$

\Omega(P)&:=\prod_{v\in P} K_v^\times \times \prod_{v \notin P} O_v^{\times}=(\mathbb{A}_K(P))^\times\\

E(P)&:=K^{\times} \cap \Omega(P)

\end{align}

Then $E(P)$ is a subgroup of $K^\{\backslash times\},$ containing all elements $\backslash xi\; \backslash in\; K^\{\backslash times\}$ satisfying $v(\backslash xi)=0$ for all $v\; \backslash notin\; P.$ Since $K^\{\backslash times\}$ is discrete in $I\_K,$ $E(P)$ is a discrete subgroup of $\backslash Omega(P)$ and with the same argument, $E(P)$ is discrete in $\backslash Omega\_1(P):=\backslash Omega(P)\backslash cap\; I\_K^1.$

An alternative definition is: $E(P)=K(P)^\{\backslash times\},$ where $K(P)$ is a subring of $K$ defined by

:$K(P):=\; K\; \backslash cap\; \backslash left(\backslash prod\_\{v\backslash in\; P\}\; K\_v\; \backslash times\; \backslash prod\_\{v\; \backslash notin\; P\}O\_v\backslash right).$

As a consequence, $K(P)$ contains all elements $\backslash xi\; \backslash in\; K,$ which fulfil $v(\backslash xi)\; \backslash geq\; 0$ for all $v\; \backslash notin\; P.$

:Lemma 1. Let The following set is finite:

::

Proof. Define

:

$W$ is compact and the set described above is the intersection of $W$ with the discrete subgroup $K^\backslash times$ in $I\_K$ and therefore finite.

:Lemma 2. Let $E$ be set of all $\backslash xi\; \backslash in\; K$ such that $|\backslash xi|\_v\; =1$ for all $v.$ Then $E\; =\; \backslash mu(K),$ the group of all roots of unity of $K.$ In particular it is finite and cyclic.

Proof. All roots of unity of $K$ have absolute value $1$ so $\backslash mu(K)\; \backslash subset\; E.$ For converse note that Lemma 1 with $c=C=1$ and any $P$ implies $E$ is finite. Moreover $E\; \backslash subset\; E(P)$ for each finite set of places $P\; \backslash supset\; P\_\backslash infty.$ Finally suppose there exists $\backslash xi\; \backslash in\; E,$ which is not a root of unity of $K.$ Then $\backslash xi^n\; \backslash neq\; 1$ for all $n\; \backslash in\; \backslash N$ contradicting the finiteness of $E.$

:Unit Theorem. $E(P)$ is the direct product of $E$ and a group isomorphic to $\backslash Z^s,$ where $s=0,$ if $P=\; \backslash emptyset$ and $s=|P|-1,$ if $P\; \backslash neq\; \backslash emptyset.$A proof can be found in {{harvnb|Weil|1967|p=78}} or in {{harvnb|Cassels|Fröhlich|1967|p=72}}.

:Dirichlet's Unit Theorem. Let $K$ be a number field. Then $O^\backslash times\backslash cong\backslash mu(K)\; \backslash times\; \backslash Z^\{r+s-1\},$ where $\backslash mu(K)$ is the finite cyclic group of all roots of unity of $K,\; r$ is the number of real embeddings of $K$ and $s$ is the number of conjugate pairs of complex embeddings of $K.$ It stands, that $[K:\backslash Q]=r+2s.$

Remark. The Unit Theorem generalises Dirichlet's Unit Theorem. To see this, let $K$ be a number field. It is already known that $E=\backslash mu(K),$ set $P=P\_\backslash infty$ and note $|P\_\backslash infty|=r+s.$

Then there is:

: $\backslash begin\{align\}$

E \times \Z^{r+s-1} = E(P_\infty)&=K^\times \cap \left(\prod_{v | \infty} K_v^\times \times \prod_{v < \infty} O_v^\times\right) \\

&\cong K^\times \cap \left( \prod_{v < \infty} O_v^\times \right) \\

&\cong O^\times

\end{align}

:Weak Approximation Theorem.A proof can be found in {{harvnb|Cassels|Fröhlich|1967|p=48}}. Let $|\backslash cdot|\_1,\; \backslash ldots,\; |\backslash cdot|\_N,$ be inequivalent valuations of $K.$ Let $K\_n$ be the completion of $K$ with respect to $|\backslash cdot|\_n.$ Embed $K$ diagonally in $K\_1\; \backslash times\; \backslash cdots\; \backslash times\; K\_N.$ Then $K$ is everywhere dense in $K\_1\; \backslash times\; \backslash cdots\; \backslash times\; K\_N.$ In other words, for each $\backslash varepsilon\; >\; 0$ and for each $(\backslash alpha\_1,\; \backslash ldots,\; \backslash alpha\_N)\; \backslash in\; K\_1\; \backslash times\; \backslash cdots\; \backslash times\; K\_N,$ there exists $\backslash xi\; \backslash in\; K,$ such that:

::

:Strong Approximation Theorem.A proof can be found in {{harvnb|Cassels|Fröhlich|1967|p=67}} Let $v\_0$ be a place of $K.$ Define

::$V:=\; \{\backslash prod\_\{v\; \backslash neq\; v\_0\}\}^\text{'}\; K\_v.$

:Then $K$ is dense in $V.$

Remark. The global field is discrete in its adele ring. The strong approximation theorem tells us that, if one place (or more) is omitted, the property of discreteness of $K$ is turned into a denseness of $K.$

:Hasse-Minkowski Theorem. A quadratic form on $K$ is zero, if and only if, the quadratic form is zero in each completion $K\_v.$

Remark. This is the Hasse principle for quadratic forms. For polynomials of degree larger than 2 the Hasse principle isn't valid in general. The idea of the Hasse principle (also known as local–global principle) is to solve a given problem of a number field $K$ by doing so in its completions $K\_v$ and then concluding on a solution in $K.$

Definition. Let $G$ be a locally compact abelian group. The character group of $G$ is the set of all characters of $G$ and is denoted by $\backslash widehat\{G\}.$ Equivalently $\backslash widehat\{G\}$ is the set of all continuous group homomorphisms from $G$ to $\backslash mathbb\{T\}:=\backslash \{z\; \backslash in\; \backslash C\; :|z|=1\backslash \}.$ Equip $\backslash widehat\{G\}$ with the topology of uniform convergence on compact subsets of $G.$ One can show that $\backslash widehat\{G\}$ is also a locally compact abelian group.

:Theorem. The adele ring is self-dual: $\backslash mathbb\{A\}\_K\backslash cong\; \backslash widehat\{\backslash mathbb\{A\}\_K\}.$

Proof. By reduction to local coordinates, it is sufficient to show each $K\_v$ is self-dual. This can be done by using a fixed character of $K\_v.$ The idea has been illustrated by showing $\backslash R$ is self-dual. Define:

:$\backslash begin\{cases\}\; e\_\backslash infty:\; \backslash R\; \backslash to\; \backslash mathbb\{T\}\; \backslash \backslash \; e\_\backslash infty(t)\; :=\backslash exp(2\backslash pi\; it)\; \backslash end\{cases\}$

Then the following map is an isomorphism which respects topologies:

:$\backslash begin\{cases\}\; \backslash phi:\; \backslash R\; \backslash to\; \backslash widehat\{\backslash R\}\; \backslash \backslash \; s\; \backslash mapsto\; \backslash begin\{cases\}\; \backslash phi\_s:\; \backslash R\; \backslash to\; \backslash mathbb\{T\}\; \backslash \backslash \; \backslash phi\_s(t)\; :=\; e\_\backslash infty(ts)\; \backslash end\{cases\}\backslash end\{cases\}$

:Theorem (algebraic and continuous duals of the adele ring).A proof can be found in {{harvnb|Weil|1967|p=66}}. Let $\backslash chi$ be a non-trivial character of $\backslash mathbb\{A\}\_K,$ which is trivial on $K.$ Let $E$ be a finite-dimensional vector-space over $K.$ Let $E^\backslash star$ and $\backslash mathbb\{A\}\_E^\backslash star$ be the algebraic duals of $E$ and $\backslash mathbb\{A\}\_E.$ Denote the topological dual of $\backslash mathbb\{A\}\_E$ by $\backslash mathbb\{A\}\_E\text{'}$ and use $\backslash langle\; \backslash cdot,\backslash cdot\; \backslash rangle$ and $[\{\backslash cdot\},\{\backslash cdot\}]$ to indicate the natural bilinear pairings on $\backslash mathbb\{A\}\_E\; \backslash times\; \backslash mathbb\{A\}\_E\text{'}$ and $\backslash mathbb\{A\}\_E\; \backslash times\; \backslash mathbb\{A\}\_E^\{\backslash star\}.$ Then the formula $\backslash langle\; e,e\text{'}\backslash rangle\; =\; \backslash chi([e,e^\backslash star])$ for all $e\; \backslash in\; \backslash mathbb\{A\}\_E$ determines an isomorphism $e^\backslash star\; \backslash mapsto\; e\text{'}$ of $\backslash mathbb\{A\}\_E^\backslash star$ onto $\backslash mathbb\{A\}\_E\text{'},$ where $e\text{'}\; \backslash in\; \backslash mathbb\{A\}\_E\text{'}$ and $e^\backslash star\; \backslash in\; \backslash mathbb\{A\}\_E^\backslash star.$ Moreover, if $e^\backslash star\; \backslash in\; \backslash mathbb\{A\}\_E^\backslash star$ fulfils $\backslash chi([e,e^\backslash star])=1$ for all $e\; \backslash in\; E,$ then $e^\backslash star\; \backslash in\; E^\backslash star.$

With the help of the characters of $\backslash mathbb\{A\}\_K,$ Fourier analysis can be done on the adele ring.For more see {{harvnb|Deitmar|2010|p=129}}. John Tate in his thesis "Fourier analysis in Number Fields and Hecke Zeta Functions"{{sfn|Cassels|Fröhlich|1967}} proved results about Dirichlet L-functions using Fourier analysis on the adele ring and the idele group. Therefore, the adele ring and the idele group have been applied to study the Riemann zeta function and more general zeta functions and the L-functions. Adelic forms of these functions can be defined and represented as integrals over the adele ring or the idele group, with respect to corresponding Haar measures. Functional equations and meromorphic continuations of these functions can be shown. For example, for all $s\; \backslash in\; \backslash C$ with $\backslash Re(s)\; >\; 1,$

:$\backslash int\_\{\backslash widehat\{\backslash Z\}\}\; |x|^s\; d^\backslash times\; x\; =\; \backslash zeta(s),$

where $d^\backslash times\; x$ is the unique Haar measure on $I\_\{\backslash Q,\backslash text\{fin\}\}$ normalised such that $\backslash widehat\{\backslash Z\}^\backslash times$ has volume one and is extended by zero to the finite adele ring. As a result, the Riemann zeta function can be written as an integral over (a subset of) the adele ring.A proof can be found {{harvnb|Deitmar|2010|p=128}}, Theorem 5.3.4. See also p. 139 for more information on Tate's thesis.

The theory of automorphic forms is a generalisation of Tate's thesis by replacing the idele group with analogous higher dimensional groups. To see this note:

:$\backslash begin\{align\}$

I_{\Q} &= \operatorname{GL} (1, \mathbb{A}_{\Q}) \\

I_{\Q}^1 &= (\operatorname{GL} (1, \mathbb{A}_\Q))^1:=\{x \in \operatorname{GL} (1, \mathbb{A}_\Q): |x|=1\} \\

\Q^{\times} &= \operatorname{GL} (1, \Q)

\end{align}

Based on these identification a natural generalisation would be to replace the idele group and the 1-idele with:

:$\backslash begin\{align\}$

I_{\Q} &\leftrightsquigarrow \operatorname{GL} (2, \mathbb{A}_{\Q}) \\

I_{\Q}^1 &\leftrightsquigarrow (\operatorname{GL} (2, \mathbb{A}_\Q))^1:=\{x \in \operatorname{GL} (2, \mathbb{A}_\Q): |\det (x) |=1\} \\

\Q &\leftrightsquigarrow \operatorname{GL} (2, \Q)

\end{align}

And finally

:$\backslash Q^\{\backslash times\}\; \backslash backslash\; I\_\{\backslash Q\}^1\; \backslash cong\; \backslash Q^\{\backslash times\}\; \backslash backslash\; I\_\{\backslash Q\}\; \backslash leftrightsquigarrow\; (\backslash operatorname\{GL\}\; (2,\; \backslash Q)\; \backslash backslash\; (\backslash operatorname\{GL\}\; (2,\; \backslash mathbb\{A\}\_\backslash Q))^1\; \backslash cong\; (\backslash operatorname\{GL\}\; (2,\; \backslash Q)Z\_\{\backslash R\})\; \backslash backslash\backslash operatorname\{GL\}\; (2,\; \backslash mathbb\{A\}\_\backslash Q),$

where $Z\_\backslash R$ is the centre of $\backslash operatorname\{GL\}\; (2,\; \backslash R).$ Then it define an automorphic form as an element of $L^2((\backslash operatorname\{GL\}\; (2,\; \backslash Q)Z\_\{\backslash R\})\; \backslash backslash\; \backslash operatorname\{GL\}\; (2,\; \backslash mathbb\{A\}\_\backslash Q)).$ In other words an automorphic form is a function on $\backslash operatorname\{GL\}\; (2,\; \backslash mathbb\{A\}\_\{\backslash Q\})$ satisfying certain algebraic and analytic conditions. For studying automorphic forms, it is important to know the representations of the group $\backslash operatorname\{GL\}\; (2,\; \backslash mathbb\{A\}\_\{\backslash Q\}).$ It is also possible to study automorphic L-functions, which can be described as integrals over $\backslash operatorname\{GL\}\; (2,\; \backslash mathbb\{A\}\_\{\backslash Q\}).$For further information see Chapters 7 and 8 in {{harvnb|Deitmar|2010}}.

Generalise even further is possible by replacing $\backslash Q$ with a number field and $\backslash operatorname\{GL\}\; (2)$ with an arbitrary reductive algebraic group.

A generalisation of Artin reciprocity law leads to the connection of representations of $\backslash operatorname\{GL\}\; (2,\; \backslash mathbb\{A\}\_K)$ and of Galois representations of $K$ (Langlands program).

The idele class group is a key object of class field theory, which describes abelian extensions of the field. The product of the local reciprocity maps in local class field theory gives a homomorphism of the idele group to the Galois group of the maximal abelian extension of the global field. The Artin reciprocity law, which is a sweeping generalisation of the Gauss quadratic reciprocity law, states that the product vanishes on the multiplicative group of the number field. Thus, the global reciprocity map of the idele class group to the abelian part of the absolute Galois group of the field will be obtained.

The self-duality of the adele ring of the function field of a curve over a finite field easily implies the Riemann–Roch theorem and the duality theory for the curve.

{{reflist}}

• {{cite book|first1=John|last1=Cassels|authorlink1=J. W. S. Cassels|first2=Albrecht|last2=Fröhlich|authorlink2=Albrecht Fröhlich|title=Algebraic number theory: proceedings of an instructional conference, organized by the London Mathematical Society, (a NATO Advanced Study Institute)|publisher=Academic Press|location=London|year=1967|volume=XVIII|isbn=978-0-12-163251-9}} 366 pages.
• {{cite book|last=Neukirch|first=Jürgen|author-link=Jürgen Neukirch|title=Algebraische Zahlentheorie, unveränd. nachdruck der 1. aufl. edn| language=de |publisher=Springer|location=Berlin|year=2007|volume=XIII|isbn=9783540375470}} 595 pages.
• {{cite book|last=Weil|first=André|author-link=André Weil|title=Basic number theory|publisher=Springer|location=Berlin; Heidelberg; New York| year=1967 |volume=XVIII|isbn=978-3-662-00048-9}} 294 pages.
• {{cite book|last=Deitmar|first=Anton|title=Automorphe Formen|language=de|publisher=Springer|location=Berlin; Heidelberg (u.a.)| year=2010 |volume=VIII |isbn=978-3-642-12389-4}} 250 pages.
• {{cite book|last=Lang|first=Serge|author-link=Serge Lang|title=Algebraic number theory, Graduate Texts in Mathematics 110| edition=2nd |publisher= Springer-Verlag|location=New York|year=1994|isbn=978-0-387-94225-4}}

• [https://mathoverflow.net/questions/78240/what-problem-do-the-adeles-solve What problem do the adeles solve?]
• [https://mathoverflow.net/questions/184737/a-good-book-on-adeles-and-ideles Some good books on adeles]

Category:Algebraic number theory

Category:Topological algebra