Idele group

In number theory, the idele group is a way of packaging the multiplicative arithmetic of a global field at all of its completions at once, so that it contains the information of unique factorization as well as the data relating to units. Formally, the idele group of a global field ${\displaystyle K}$ is the restricted direct product ${\displaystyle \mathbb {A} _{K}^{\times }=\prod _{v}'K_{v}^{\times }}$ of the multiplicative groups of the completions of ${\displaystyle K}$, taken with respect to the unit groups ${\displaystyle {\mathcal {O}}_{v}^{\times }}$ at the non-archimedean places. Equivalently, it is the group of invertible elements of the adele ring ${\displaystyle \mathbb {A} _{K}}$, equipped with a topology finer than the subspace topology inherited from ${\displaystyle \mathbb {A} _{K}}$.

The quotient ${\displaystyle C_{K}=\mathbb {A} _{K}^{\times }/K^{\times }}$ is the idele class group. Ideles and idele class groups are used in class field theory. They were exploited by John Tate in his thesis to formulate global zeta and ${\displaystyle L}$-functions and Hecke characters.

Let ${\displaystyle K}$ be a global field, and let ${\displaystyle v}$ run over the places of ${\displaystyle K}$. For each place ${\displaystyle v}$, let ${\displaystyle K_{v}}$ denote the completion of ${\displaystyle K}$ at ${\displaystyle v}$. If ${\displaystyle v}$ is non-archimedean, let ${\displaystyle {\mathcal {O}}_{v}}$ be the corresponding valuation ring and let ${\displaystyle {\mathcal {O}}_{v}^{\times }}$ be its group of units.

The idele group of ${\displaystyle K}$, usually denoted ${\displaystyle \mathbb {A} _{K}^{\times }}$ or ${\displaystyle I_{K}}$, is the restricted product

${\displaystyle \mathbb {A} _{K}^{\times }=\prod _{v}'K_{v}^{\times }}$

of the groups ${\displaystyle K_{v}^{\times }}$, taken with respect to the subgroups ${\displaystyle {\mathcal {O}}_{v}^{\times }}$ at the non-archimedean places. Thus an idele is a family

${\displaystyle x=(x_{v})_{v},\qquad x_{v}\in K_{v}^{\times },}$

such that

${\displaystyle x_{v}\in {\mathcal {O}}_{v}^{\times }}$

for all but finitely many non-archimedean places ${\displaystyle v}$. Multiplication is defined componentwise.¹²

Equivalently, the idele group is the group of invertible elements of the adele ring ${\displaystyle \mathbb {A} _{K}}$. However, its topology is not the subspace topology inherited from ${\displaystyle \mathbb {A} _{K}}$; it is the restricted product topology, or equivalently the topology induced by the embedding

${\displaystyle \mathbb {A} _{K}^{\times }\longrightarrow \mathbb {A} _{K}\times \mathbb {A} _{K},\qquad x\mapsto (x,x^{-1}).}$

The multiplicative group ${\displaystyle K^{\times }}$ embeds diagonally in ${\displaystyle \mathbb {A} _{K}^{\times }}$. The quotient

${\displaystyle C_{K}=\mathbb {A} _{K}^{\times }/K^{\times }}$

is called the idele class group of ${\displaystyle K}$.

The idele group may be viewed as a topological refinement of the group of fractional ideals of a number field. If ${\displaystyle K}$ is a number field with ring of integers ${\displaystyle {\mathcal {O}}_{K}}$, every nonzero fractional ideal has a unique factorization

${\displaystyle {\mathfrak {a}}=\prod _{\mathfrak {p}}{\mathfrak {p}}^{n_{\mathfrak {p}}},}$

where ${\displaystyle {\mathfrak {p}}}$ runs over the nonzero prime ideals of ${\displaystyle {\mathcal {O}}_{K}}$ and all but finitely many integers ${\displaystyle n_{\mathfrak {p}}}$ are zero. Thus the group of fractional ideals records, for each finite place of ${\displaystyle K}$, an integral valuation.

An idele records similar local valuation data, but with additional local information. For an idele ${\displaystyle x=(x_{v})_{v}}$, the component ${\displaystyle x_{\mathfrak {p}}\in K_{\mathfrak {p}}^{\times }}$ at a finite place determines an integer ${\displaystyle v_{\mathfrak {p}}(x_{\mathfrak {p}})}$. Since ${\displaystyle x_{\mathfrak {p}}}$ is a unit for all but finitely many ${\displaystyle {\mathfrak {p}}}$, these integers define a fractional ideal

${\displaystyle (x)_{\mathrm {fin} }=\prod _{\mathfrak {p}}{\mathfrak {p}}^{v_{\mathfrak {p}}(x_{\mathfrak {p}})}.}$

This gives a surjective homomorphism from the idele group to the group of fractional ideals. The diagonal embedding ${\displaystyle K^{\times }\hookrightarrow \mathbb {A} _{K}^{\times }}$ sends an element ${\displaystyle a\in K^{\times }}$ to the principal idele whose associated fractional ideal is the principal ideal ${\displaystyle (a)}$. Consequently, passing to quotients gives a natural surjection

${\displaystyle \mathbb {A} _{K}^{\times }/K^{\times }\longrightarrow \operatorname {Cl} (K),}$

from the idele class group to the ordinary ideal class group.³

Thus the idele class group enlarges the ideal class group. It extends the finite-prime data measured by fractional ideals with the unit groups at finite places and the multiplicative groups at the archimedean places. This additional topological information is important in class field theory and in the theory of Hecke characters, where characters of the idele class group replace characters defined only on ideal class groups or ray class groups.

Although the idele group ${\displaystyle \mathbb {A} _{K}^{\times }}$ is the group of invertible elements of the adele ring ${\displaystyle \mathbb {A} _{K}}$, it is not usually equipped with the subspace topology inherited from ${\displaystyle \mathbb {A} _{K}}$. With the subspace topology, inversion need not be continuous. Instead, ${\displaystyle \mathbb {A} _{K}^{\times }}$ is given the restricted product topology

${\displaystyle \mathbb {A} _{K}^{\times }=\prod _{v}'K_{v}^{\times },}$

where the restricted product is taken with respect to the compact open subgroups ${\displaystyle {\mathcal {O}}_{v}^{\times }}$ at the non-archimedean places. A basis of open neighbourhoods of the identity is given by products

${\displaystyle \prod _{v}U_{v},}$

where ${\displaystyle U_{v}}$ is an open neighbourhood of ${\displaystyle 1}$ in ${\displaystyle K_{v}^{\times }}$ and ${\displaystyle U_{v}={\mathcal {O}}_{v}^{\times }}$ for all but finitely many non-archimedean places ${\displaystyle v}$. Equivalently, this is the topology induced by the embedding

${\displaystyle \mathbb {A} _{K}^{\times }\longrightarrow \mathbb {A} _{K}\times \mathbb {A} _{K},\qquad x\mapsto (x,x^{-1}).}$

With this topology, ${\displaystyle \mathbb {A} _{K}^{\times }}$ is a locally compact topological group.²¹

Since the idele group is locally compact, it has a Haar measure, usually denoted ${\displaystyle d^{\times }x}$. This measure is obtained as a product of local multiplicative Haar measures on the groups ${\displaystyle K_{v}^{\times }}$. At a non-archimedean place ${\displaystyle v}$, the local measure is commonly normalized so that

${\displaystyle \operatorname {vol} ({\mathcal {O}}_{v}^{\times })=1.}$

At the real place, a standard multiplicative Haar measure on ${\displaystyle \mathbb {R} ^{\times }}$ is

${\displaystyle d^{\times }x={\frac {dx}{|x|}},}$

up to multiplication by a positive constant; analogous normalizations are used at complex places. These local choices combine to give a multiplicative Haar measure on ${\displaystyle \mathbb {A} _{K}^{\times }}$. Such measures are used in harmonic analysis on the ideles, especially in Tate's thesis and in the analytic theory of Hecke ${\displaystyle L}$-functions.⁴⁵

The idele group carries a homomorphism, usually called the idele norm or module into the positive reals. Choose the standard normalized absolute value ${\displaystyle |\cdot |_{v}}$ on each completion ${\displaystyle K_{v}}$: for a non-archimedean place ${\displaystyle v}$, it is normalized so that $${\displaystyle |\varpi _{v}|_{v}=q_{v}^{-1},}$$ where ${\displaystyle \varpi _{v}}$ is a uniformizer and ${\displaystyle q_{v}}$ is the size of the residue field. At the archimedean places one uses the usual normalized absolute values, with the complex absolute value taken squared. For an idele ${\displaystyle x=(x_{v})_{v}}$, define $${\displaystyle |x|_{\mathbb {A} }=\prod _{v}|x_{v}|_{v}.}$$

This product is finite, since ${\displaystyle x_{v}\in {\mathcal {O}}_{v}^{\times }}$ for all but finitely many non-archimedean places, and hence ${\displaystyle |x_{v}|_{v}=1}$ for all but finitely many ${\displaystyle v}$. Thus $${\displaystyle |\cdot |_{\mathbb {A} }:\mathbb {A} _{K}^{\times }\to \mathbb {R} _{>0}}$$ is a continuous group homomorphism.¹²

The norm-one ideles are the elements in the kernel of this homomorphism: $${\displaystyle \mathbb {A} _{K}^{1}=\{x\in \mathbb {A} _{K}^{\times }:|x|_{\mathbb {A} }=1\}.}$$

By the product formula for global fields, every element of ${\displaystyle K^{\times }}$, embedded diagonally in ${\displaystyle \mathbb {A} _{K}^{\times }}$, has idele norm one. Hence

$${\displaystyle K^{\times }\subset \mathbb {A} _{K}^{1}.}$$

The quotient $${\displaystyle C_{K}^{1}=\mathbb {A} _{K}^{1}/K^{\times }}$$ is called the group of norm-one idele classes. It is a compact group.²⁶

The idele norm descends to a homomorphism on the idele class group, $${\displaystyle |\cdot |_{\mathbb {A} }:C_{K}=\mathbb {A} _{K}^{\times }/K^{\times }\to \mathbb {R} _{>0},}$$ whose kernel is ${\displaystyle C_{K}^{1}}$. For number fields this gives an exact sequence $${\displaystyle 1\longrightarrow C_{K}^{1}\longrightarrow C_{K}\xrightarrow {|\cdot |_{\mathbb {A} }} \mathbb {R} _{>0}\longrightarrow 1.}$$

Thus the idele class group is not compact in the number field case, but its norm-one subgroup modulo ${\displaystyle K^{\times }}$ is compact. This compactness is one of the idelic forms of the finiteness of the ideal class group together with the structure theorem for units.¹²

For number fields, the idele norm is surjective onto ${\displaystyle \mathbb {R} _{>0}}$, and the above exact sequence splits after choosing a positive archimedean component. Thus ${\displaystyle C_{K}}$ is, non-canonically or after such a choice, a product of the compact group ${\displaystyle C_{K}^{1}}$ with ${\displaystyle \mathbb {R} _{>0}}$. For global function fields, the image of the idele norm is instead a discrete subgroup of ${\displaystyle \mathbb {R} _{>0}}$, so the corresponding quotient is discrete and isomorphic to an infinite cyclic group.²¹

Let ${\displaystyle L/K}$ be a finite extension of global fields. For each place ${\displaystyle v}$ of ${\displaystyle K}$ and each place ${\displaystyle w}$ of ${\displaystyle L}$ lying above ${\displaystyle v}$, there is a local norm map

${\displaystyle N_{L_{w}/K_{v}}:L_{w}^{\times }\longrightarrow K_{v}^{\times }.}$

These local norm maps combine to give a continuous homomorphism on idele groups

${\displaystyle N_{L/K}:\mathbb {A} _{L}^{\times }\longrightarrow \mathbb {A} _{K}^{\times }.}$

If ${\displaystyle y=(y_{w})_{w}\in \mathbb {A} _{L}^{\times }}$, then the ${\displaystyle v}$-component of ${\displaystyle N_{L/K}(y)}$ is

${\displaystyle \left(N_{L/K}(y)\right)_{v}=\prod _{w\mid v}N_{L_{w}/K_{v}}(y_{w}).}$

This product is finite for each fixed ${\displaystyle v}$. Moreover, for all but finitely many non-archimedean places ${\displaystyle w}$, the component ${\displaystyle y_{w}}$ lies in ${\displaystyle {\mathcal {O}}_{w}^{\times }}$, and its local norm lies in ${\displaystyle {\mathcal {O}}_{v}^{\times }}$. Hence ${\displaystyle N_{L/K}(y)}$ is again an idele of ${\displaystyle K}$. The continuity follows from the continuity of the local norm maps and from the restricted product topology.¹²

The norm map is compatible with principal ideles. If ${\displaystyle a\in L^{\times }}$ is embedded diagonally in ${\displaystyle \mathbb {A} _{L}^{\times }}$, then

${\displaystyle N_{L/K}(a)}$

is the principal idele of ${\displaystyle K}$ associated with the field norm ${\displaystyle N_{L/K}(a)\in K^{\times }}$. Consequently, the idele norm descends to a continuous homomorphism on idele class groups,

${\displaystyle N_{L/K}:C_{L}\longrightarrow C_{K},}$

where ${\displaystyle C_{L}=\mathbb {A} _{L}^{\times }/L^{\times }}$ and ${\displaystyle C_{K}=\mathbb {A} _{K}^{\times }/K^{\times }}$.

The embedding of ${\displaystyle K}$ into ${\displaystyle L}$ also gives a natural homomorphism

${\displaystyle \mathbb {A} _{K}^{\times }\longrightarrow \mathbb {A} _{L}^{\times }.}$

Explicitly, an idele ${\displaystyle x=(x_{v})_{v}}$ of ${\displaystyle K}$ is sent to the idele whose component at ${\displaystyle w\mid v}$ is the image of ${\displaystyle x_{v}}$ in ${\displaystyle L_{w}^{\times }}$. Under this embedding,

${\displaystyle N_{L/K}(x)=x^{[L:K]},}$

where the power is taken componentwise. This follows from the identity

${\displaystyle \prod _{w\mid v}N_{L_{w}/K_{v}}(x_{v})=x_{v}^{\sum _{w\mid v}[L_{w}:K_{v}]}=x_{v}^{[L:K]}.}$

The field-extension norm should be distinguished from the idele norm or module ${\displaystyle |x|_{\mathbb {A} }}$. They are nevertheless compatible: with the standard normalized absolute values,

${\displaystyle |N_{L/K}(y)|_{\mathbb {A} _{K}}=|y|_{\mathbb {A} _{L}}.}$

In particular, ${\displaystyle N_{L/K}}$ maps the norm-one idele group ${\displaystyle \mathbb {A} _{L}^{1}}$ into ${\displaystyle \mathbb {A} _{K}^{1}}$ and induces a homomorphism

${\displaystyle C_{L}^{1}\longrightarrow C_{K}^{1}.}$

In global class field theory, the image ${\displaystyle N_{L/K}(C_{L})}$ is called the norm subgroup of ${\displaystyle C_{K}}$. For a finite abelian extension ${\displaystyle L/K}$, the global Artin reciprocity map identifies the quotient

${\displaystyle C_{K}/N_{L/K}(C_{L})}$

with the Galois group ${\displaystyle \operatorname {Gal} (L/K)}$, up to the usual convention concerning arithmetic or geometric Frobenius.³⁶⁷

For ${\displaystyle K=\mathbb {Q} }$, the finite adele ring is

${\displaystyle \mathbb {A} _{\mathbb {Q} ,\mathrm {fin} }=\prod _{p}'\mathbb {Q} _{p},}$

and the finite integral adeles are

${\displaystyle {\widehat {\mathbb {Z} }}=\prod _{p}\mathbb {Z} _{p}.}$

The finite ideles are

${\displaystyle \mathbb {A} _{\mathbb {Q} ,\mathrm {fin} }^{\times }=\prod _{p}'\mathbb {Q} _{p}^{\times },}$

where the restricted product is taken with respect to ${\displaystyle \mathbb {Z} _{p}^{\times }}$. The idele group of ${\displaystyle \mathbb {Q} }$ is

${\displaystyle \mathbb {A} _{\mathbb {Q} }^{\times }=\mathbb {A} _{\mathbb {Q} ,\mathrm {fin} }^{\times }\times \mathbb {R} ^{\times }.}$

Every idele class has a representative of the form

${\displaystyle (u,t)\in {\widehat {\mathbb {Z} }}^{\times }\times \mathbb {R} _{>0}.}$

Indeed, multiplying by a rational number changes the finite valuations and can be used to make all finite components ${\displaystyle p}$-adic units; the remaining positive real factor records the idele norm. Thus

${\displaystyle \mathbb {A} _{\mathbb {Q} }^{\times }/\mathbb {Q} ^{\times }\cong {\widehat {\mathbb {Z} }}^{\times }\times \mathbb {R} _{>0}.}$

Similarly, the norm-one idele classes are

${\displaystyle \mathbb {A} _{\mathbb {Q} }^{1}/\mathbb {Q} ^{\times }\cong {\widehat {\mathbb {Z} }}^{\times }.}$

This reflects the fact that ${\displaystyle \mathbb {Q} }$ has trivial ideal class group: the remaining finite part of the idele class group comes from the local unit groups ${\displaystyle \mathbb {Z} _{p}^{\times }}$.

The idele class group yields a formulation of class field theory. Global class field theory describes the abelian extensions of a global field ${\displaystyle K}$ in terms of topological quotients of ${\displaystyle C_{K}=\mathbb {A} _{K}^{\times }/K^{\times }.}$

The main result is the global Artin reciprocity law. In one formulation, for every finite abelian extension ${\displaystyle L/K}$ there is a canonical reciprocity homomorphism $${\displaystyle \theta _{L/K}:C_{K}\longrightarrow \operatorname {Gal} (L/K),}$$ whose kernel is the norm subgroup ${\displaystyle N_{L/K}(C_{L})\subset C_{K}.}$

The reciprocity homomorphism induces an isomorphism $${\displaystyle C_{K}/N_{L/K}(C_{L})\cong \operatorname {Gal} (L/K),}$$ up to a conventional choice of arithmetic or geometric Frobenius automorphism.³⁶⁷

Thus, finite abelian extensions of ${\displaystyle K}$ correspond to open subgroups of finite index in the idele class group. Under this correspondence, an extension ${\displaystyle L/K}$ is associated with the subgroup ${\displaystyle N_{L/K}(C_{L})}$. This replaces the older formulation of class field theory in terms of ideal class groups, ray class groups, and congruence conditions by a topological statement about quotients of ${\displaystyle C_{K}}$.³⁷

The idelic formulation also incorporates the local reciprocity maps of local class field theory: for each place ${\displaystyle v}$ of ${\displaystyle K}$, local class field theory relates ${\displaystyle K_{v}^{\times }}$ to the abelianized Galois group of ${\displaystyle K_{v}}$. The global reciprocity map is compatible with these local maps through the embedding of each local multiplicative group into the idele group. At an unramified finite place, a uniformizer maps to a Frobenius element, with the precise inverse depending on the convention used for the Artin map.³⁷

Classical ideal-theoretic class field theory is then a special case. Quotients of the idele class group by certain subgroups recover ray class groups, and the corresponding abelian extensions are the ray class fields. In particular, the Hilbert class field is obtained from the quotient associated with the ordinary ideal class group, packaging the relation between ideles, fractional ideals, and ideal classes.³⁶

For the maximal abelian extension ${\displaystyle K^{\mathrm {ab} }}$, the finite-level reciprocity maps are compatible as ${\displaystyle L}$ varies over finite abelian extensions of ${\displaystyle K}$. They combine into a global reciprocity map from the idele class group to ${\displaystyle \operatorname {Gal} (K^{\mathrm {ab} }/K)}$. Thus the abelianized absolute Galois group of ${\displaystyle K}$ is described by the system of finite quotients of the idele class group.³⁷

A Hecke character of a global field ${\displaystyle K}$ can be described as a continuous homomorphism

$${\displaystyle \chi :\mathbb {A} _{K}^{\times }/K^{\times }\to \mathbb {C} ^{\times },}$$

or equivalently as a continuous character of the idele group ${\displaystyle \mathbb {A} _{K}^{\times }}$ that is trivial on the diagonally embedded subgroup ${\displaystyle K^{\times }}$. Such characters are the automorphic characters of ${\displaystyle \operatorname {GL} _{1}(\mathbb {A} _{K})}$.

Writing an idele as ${\displaystyle x=(x_{v})_{v}}$, a Hecke character decomposes into local characters

$${\displaystyle \chi _{v}:K_{v}^{\times }\to \mathbb {C} ^{\times },}$$

with

$${\displaystyle \chi (x)=\prod _{v}\chi _{v}(x_{v}).}$$

For all but finitely many non-archimedean places ${\displaystyle v}$, the local character ${\displaystyle \chi _{v}}$ is unramified, meaning that it is trivial on ${\displaystyle {\mathcal {O}}_{v}^{\times }}$. At such a place its value is determined by ${\displaystyle \chi _{v}(\varpi _{v})}$, where ${\displaystyle \varpi _{v}}$ is a uniformizer of ${\displaystyle K_{v}}$.

Associated to a Hecke character is a global ${\displaystyle L}$-function, defined for suitable ${\displaystyle s}$ by an Euler product

$${\displaystyle L(s,\chi )=\prod _{v}L_{v}(s,\chi _{v}).}$$

At an unramified non-archimedean place ${\displaystyle v}$, the local factor has the form

$${\displaystyle L_{v}(s,\chi _{v})=\left(1-\chi _{v}(\varpi _{v})q_{v}^{-s}\right)^{-1},}$$

where ${\displaystyle q_{v}}$ is the size of the residue field. The remaining finitely many finite places give ramified local factors, and the archimedean places contribute gamma factors. These local factors combine to form the completed Hecke ${\displaystyle L}$-function.⁴⁸

Classical Dirichlet characters and ideal class characters occur as special cases. For example, over ${\displaystyle \mathbb {Q} }$, Dirichlet characters can be interpreted as finite-order Hecke characters with prescribed finite conductors. More generally, ray class characters of a number field can be realized as finite-order characters of quotients of the idele class group.

Hecke ${\displaystyle L}$-functions are among the basic examples of automorphic ${\displaystyle L}$-functions. In Tate's thesis, the analytic continuation and functional equation of these ${\displaystyle L}$-functions are obtained by harmonic analysis on the adele ring and the idele group. This approach recovers the analytic theory of Dirichlet ${\displaystyle L}$-functions and Hecke's original ${\displaystyle L}$-series, while also explaining their local-global factorization in terms of the product structure of the ideles.⁴⁹

For a number field ${\displaystyle K}$, the idele group refines the ordinary ideal-theoretic arithmetic of ${\displaystyle K}$. Let ${\displaystyle {\mathcal {O}}_{K}}$ be the ring of integers of ${\displaystyle K}$, let ${\displaystyle J_{K}}$ be the group of nonzero fractional ideals of ${\displaystyle K}$, and let

${\displaystyle {\widehat {\mathcal {O}}}_{K}=\prod _{\mathfrak {p}}{\mathcal {O}}_{\mathfrak {p}}}$

be the profinite completion of ${\displaystyle {\mathcal {O}}_{K}}$, where ${\displaystyle {\mathfrak {p}}}$ runs over the nonzero prime ideals of ${\displaystyle {\mathcal {O}}_{K}}$. Its group of units is

${\displaystyle {\widehat {\mathcal {O}}}_{K}^{\times }=\prod _{\mathfrak {p}}{\mathcal {O}}_{\mathfrak {p}}^{\times }.}$

Let ${\displaystyle I_{K,\mathrm {fin} }}$ denote the finite idele group,

${\displaystyle I_{K,\mathrm {fin} }=\prod _{\mathfrak {p}}'K_{\mathfrak {p}}^{\times }.}$

There is a natural surjective homomorphism

${\displaystyle I_{K,\mathrm {fin} }\longrightarrow J_{K}}$

defined by

${\displaystyle x=(x_{\mathfrak {p}})_{\mathfrak {p}}\longmapsto \prod _{\mathfrak {p}}{\mathfrak {p}}^{v_{\mathfrak {p}}(x_{\mathfrak {p}})},}$

where ${\displaystyle v_{\mathfrak {p}}}$ is the normalized additive valuation at ${\displaystyle {\mathfrak {p}}}$. The product is finite because ${\displaystyle x_{\mathfrak {p}}\in {\mathcal {O}}_{\mathfrak {p}}^{\times }}$ for all but finitely many ${\displaystyle {\mathfrak {p}}}$. The kernel of this homomorphism is exactly ${\displaystyle {\widehat {\mathcal {O}}}_{K}^{\times }}$. Hence

${\displaystyle I_{K,\mathrm {fin} }/{\widehat {\mathcal {O}}}_{K}^{\times }\cong J_{K}.}$

This identifies the group of fractional ideals with the quotient of the finite idele group obtained by forgetting the local unit components.¹⁶

The diagonal embedding ${\displaystyle K^{\times }\hookrightarrow I_{K,\mathrm {fin} }}$ is compatible with principal ideals. If ${\displaystyle a\in K^{\times }}$, then the finite idele whose components are all equal to ${\displaystyle a}$ maps to the principal fractional ideal ${\displaystyle (a)}$. Therefore the preceding homomorphism descends to a quotient map from finite idele classes to ideal classes. In particular,

${\displaystyle \operatorname {Cl} (K)\cong I_{K,\mathrm {fin} }/K^{\times }{\widehat {\mathcal {O}}}_{K}^{\times }.}$

Equivalently, using the full idele group,

${\displaystyle \operatorname {Cl} (K)\cong \mathbb {A} _{K}^{\times }{\big /}K^{\times }\left({\widehat {\mathcal {O}}}_{K}^{\times }\times \prod _{v\mid \infty }K_{v}^{\times }\right).}$

Thus the ordinary ideal class group is obtained from the idele class group by quotienting out the finite local unit groups and the archimedean multiplicative factors.

The same construction gives a useful way to view why ideles contain more information than ideals. Passing from an idele ${\displaystyle x=(x_{v})_{v}}$ to the associated fractional ideal records only the valuations ${\displaystyle v_{\mathfrak {p}}(x_{\mathfrak {p}})}$ at the finite places. It discards the unit components in ${\displaystyle {\mathcal {O}}_{\mathfrak {p}}^{\times }}$ and also discards the archimedean components. These extra local and topological data are precisely what make the idele class group suitable for class field theory and for the theory of Hecke characters.

A proof sketch is as follows. For each finite prime ${\displaystyle {\mathfrak {p}}}$, choose a uniformizer ${\displaystyle \varpi _{\mathfrak {p}}}$ of ${\displaystyle K_{\mathfrak {p}}}$. Every element of ${\displaystyle K_{\mathfrak {p}}^{\times }}$ can be written as ${\displaystyle \varpi _{\mathfrak {p}}^{n}u}$, with ${\displaystyle n\in \mathbb {Z} }$ and ${\displaystyle u\in {\mathcal {O}}_{\mathfrak {p}}^{\times }}$. Hence the valuation map records exactly the exponent of ${\displaystyle {\mathfrak {p}}}$. Since an idele is a unit at almost all finite places, only finitely many exponents are nonzero, so the formula above defines a fractional ideal. The kernel consists exactly of those finite ideles with all valuations zero, namely ${\displaystyle {\widehat {\mathcal {O}}}_{K}^{\times }}$. Surjectivity follows because any fractional ideal ${\displaystyle \prod _{\mathfrak {p}}{\mathfrak {p}}^{n_{\mathfrak {p}}}}$ is represented by the finite idele whose ${\displaystyle {\mathfrak {p}}}$-component is ${\displaystyle \varpi _{\mathfrak {p}}^{n_{\mathfrak {p}}}}$ for the finitely many primes appearing in the product and is ${\displaystyle 1}$ elsewhere. Finally, quotienting by the diagonal image of ${\displaystyle K^{\times }}$ identifies principal fractional ideals with principal ideles, giving the ideal class group.

The following standard structural facts give equivalent descriptions of the idele topology, related subgroups, and some compactness and decomposition results used in the arithmetic theory of ideles.

The topology on ${\displaystyle \mathbb {A} _{K}^{\times }}$ can be described by a general construction for unit groups of topological rings. Let ${\displaystyle R}$ be a topological ring. Define

${\displaystyle {\begin{cases}\iota :R^{\times }\to R\times R\\x\mapsto (x,x^{-1}).\end{cases}}}$

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

Proof.

Since ${\displaystyle R}$ is a topological ring, it is sufficient to show that the inverse map is continuous. Let ${\displaystyle U\subset R^{\times }}$ be open. Then ${\displaystyle U\times U^{-1}\subset R\times R}$ is open. It is necessary to show that ${\displaystyle U^{-1}\subset R^{\times }}$ is open, or equivalently that

${\displaystyle U^{-1}\times (U^{-1})^{-1}=U^{-1}\times U\subset R\times R}$

is open. But this is the same condition applied to ${\displaystyle U^{-1}}$. The idele group is equipped with this topology.

The subset topology inherited from ${\displaystyle \mathbb {A} _{K}}$ is not a suitable candidate in general, since the group of units of a topological ring equipped with the subset topology may not be a topological group. For example, the inverse map in ${\displaystyle \mathbb {A} _{\mathbb {Q} }}$ is not continuous. The sequence

${\displaystyle {\begin{aligned}x_{1}&=(2,1,\ldots )\\x_{2}&=(1,3,1,\ldots )\\x_{3}&=(1,1,5,1,\ldots )\\&\vdots \end{aligned}}}$

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

${\displaystyle U=\prod _{p\leq N}U_{p}\times \prod _{p>N}\mathbb {Z} _{p}.}$

Since ${\displaystyle (x_{n})_{p}-1\in \mathbb {Z} _{p}}$ for all ${\displaystyle p}$, it follows that ${\displaystyle x_{n}-1\in U}$ for ${\displaystyle n}$ large enough. However, the inverses of this sequence do not converge to ${\displaystyle 1}$ in ${\displaystyle \mathbb {A} _{\mathbb {Q} }}$.

For ${\displaystyle S}$ a subset of places of ${\displaystyle K}$, set

${\displaystyle I_{K,S}:=\mathbb {A} _{K,S}^{\times },\qquad I_{K}^{S}:=(\mathbb {A} _{K}^{S})^{\times }.}$

The following identities of topological groups hold:

${\displaystyle {\begin{aligned}I_{K,S}&={\prod _{v\in S}}^{'}K_{v}^{\times },\\I_{K}^{S}&={\prod _{v\notin S}}^{'}K_{v}^{\times },\\I_{K}&={\prod _{v}}^{'}K_{v}^{\times }.\end{aligned}}}$

Here the restricted product has the restricted product topology, generated by restricted open rectangles of the form

${\displaystyle \prod _{v\in E}U_{v}\times \prod _{v\notin E}{\mathcal {O}}_{v}^{\times },}$

where ${\displaystyle E}$ is a finite subset of the set of all places and ${\displaystyle U_{v}\subset K_{v}^{\times }}$ are open sets.

Proof.

It suffices to prove the identity for ${\displaystyle I_{K}}$; the other two follow similarly. First show the two sets are equal:

${\displaystyle {\begin{aligned}I_{K}&=\{x=(x_{v})_{v}\in \mathbb {A} _{K}:\exists y=(y_{v})_{v}\in \mathbb {A} _{K}:xy=1\}\\&=\{x=(x_{v})_{v}\in \mathbb {A} _{K}:\exists y=(y_{v})_{v}\in \mathbb {A} _{K}:x_{v}y_{v}=1\quad \forall v\}\\&=\{x=(x_{v})_{v}:x_{v}\in K_{v}^{\times }\ \forall v{\text{ and }}x_{v}\in {\mathcal {O}}_{v}^{\times }{\text{ for almost all }}v\}\\&={\prod _{v}}'K_{v}^{\times }.\end{aligned}}}$

In going from the second line to the third, ${\displaystyle x}$ as well as ${\displaystyle x^{-1}=y}$ have to be in ${\displaystyle \mathbb {A} _{K}}$, meaning ${\displaystyle x_{v}\in {\mathcal {O}}_{v}}$ for almost all ${\displaystyle v}$ and ${\displaystyle x_{v}^{-1}\in {\mathcal {O}}_{v}}$ for almost all ${\displaystyle v}$. Therefore ${\displaystyle x_{v}\in {\mathcal {O}}_{v}^{\times }}$ for almost all ${\displaystyle v}$.

Now the topology on the left-hand side equals the topology on the right-hand side. Every open restricted rectangle is open in the topology of the idele group. Conversely, for a given ${\displaystyle U\subset I_{K}}$ open in the topology of the idele group, meaning that ${\displaystyle U\times U^{-1}\subset \mathbb {A} _{K}\times \mathbb {A} _{K}}$ is open, for each ${\displaystyle u\in U}$ there exists an open restricted rectangle contained in ${\displaystyle U}$ and containing ${\displaystyle u}$. Therefore ${\displaystyle U}$ is the union of all these restricted open rectangles and is open in the restricted product topology.

For each set of places ${\displaystyle S}$, ${\displaystyle I_{K,S}}$ is a locally compact topological group. The local compactness follows from the description of ${\displaystyle I_{K,S}}$ as a restricted product, and the topological group property follows from the preceding discussion on the group of units of a topological ring.

A neighbourhood system of ${\displaystyle 1\in I_{K}}$ is given by all sets of the form

${\displaystyle \prod _{v}U_{v},}$

where ${\displaystyle U_{v}}$ is a neighbourhood of ${\displaystyle 1\in K_{v}^{\times }}$ and ${\displaystyle U_{v}={\mathcal {O}}_{v}^{\times }}$ for almost all ${\displaystyle v}$.

Let ${\displaystyle L/K}$ be a finite extension. Then

${\displaystyle I_{L}={\prod _{w}}'L_{w}^{\times },}$

where the restricted product is with respect to the unit groups ${\displaystyle {\mathcal {O}}_{w}^{\times }}$.

There is a canonical embedding of ${\displaystyle I_{K}}$ in ${\displaystyle I_{L}}$. Map ${\displaystyle a=(a_{v})_{v}\in I_{K}}$ to ${\displaystyle a'=(a'_{w})_{w}\in I_{L}}$ with the property

${\displaystyle a'_{w}=a_{v}\in K_{v}^{\times }\subset L_{w}^{\times }}$

for ${\displaystyle w\mid v}$. Therefore ${\displaystyle I_{K}}$ can be seen as a subgroup of ${\displaystyle I_{L}}$. An element ${\displaystyle a=(a_{w})_{w}\in I_{L}}$ is in this subgroup if and only if its components satisfy the following properties: ${\displaystyle a_{w}\in K_{v}^{\times }}$ for ${\displaystyle w\mid v}$, and ${\displaystyle a_{w}=a_{w'}}$ for ${\displaystyle w\mid v}$ and ${\displaystyle w'\mid v}$ over the same place ${\displaystyle v}$ of ${\displaystyle K}$.

The embedding ${\displaystyle I_{K}\to I_{L}}$ induces an injective map

${\displaystyle {\begin{cases}C_{K}\to C_{L},\\\alpha K^{\times }\mapsto \alpha L^{\times }.\end{cases}}}$

There is a natural embedding of ${\displaystyle K^{\times }}$ into ${\displaystyle I_{K}}$ given by the diagonal map

${\displaystyle a\mapsto (a,a,a,\ldots ).}$

Since ${\displaystyle K^{\times }}$ is a subset of ${\displaystyle K_{v}^{\times }}$ for all ${\displaystyle v}$, the embedding is well-defined and injective. In analogy to the ideal class group, the elements of ${\displaystyle K^{\times }}$ in ${\displaystyle I_{K}}$ are called principal ideles.

The subgroup ${\displaystyle K^{\times }}$ is closed and discrete in ${\displaystyle I_{K}}$. Therefore

${\displaystyle C_{K}=I_{K}/K^{\times }}$

is a locally compact topological group and a Hausdorff space.

More generally, in the adelic algebra setting described below, ${\displaystyle A^{\times }}$ is a discrete subgroup of ${\displaystyle \mathbb {A} _{A}^{\times }}$.

For ${\displaystyle \alpha =(\alpha _{v})_{v}\in I_{K}}$, define

${\displaystyle |\alpha |:=\prod _{v}|\alpha _{v}|_{v}.}$

Since ${\displaystyle \alpha }$ is an idele, this product is finite and therefore well-defined. The set of norm-one ideles is

${\displaystyle I_{K}^{1}:=\{x\in I_{K}:|x|=1\}=\ker(|\cdot |).}$

The subgroup ${\displaystyle I_{K}^{1}}$ is a closed subgroup of ${\displaystyle I_{K}}$. The ${\displaystyle \mathbb {A} _{K}}$-topology on ${\displaystyle I_{K}^{1}}$ equals the subspace topology of ${\displaystyle I_{K}}$ on ${\displaystyle I_{K}^{1}}$.¹⁰

The product formula states that

${\displaystyle |k|=1}$

for all ${\displaystyle k\in K^{\times }}$.

Proof.

For number fields, the case of global function fields being similar, let ${\displaystyle K}$ be a number field and ${\displaystyle a\in K^{\times }}$. It has to be shown that

${\displaystyle \prod _{v}|a|_{v}=1.}$

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

${\displaystyle {\begin{aligned}\prod _{v}|a|_{v}&=\prod _{p\leq \infty }\prod _{v\mid p}|a|_{v}\\&=\prod _{p\leq \infty }\prod _{v\mid p}|N_{K_{v}/\mathbb {Q} _{p}}(a)|_{p}\\&=\prod _{p\leq \infty }|N_{K/\mathbb {Q} }(a)|_{p}.\end{aligned}}}$

In going from the first line to the second, the identity

${\displaystyle |a|_{w}=|N_{L_{w}/K_{v}}(a)|_{v}}$

is used, where ${\displaystyle v}$ is a place of ${\displaystyle K}$ and ${\displaystyle w}$ is a place of ${\displaystyle L}$ lying above ${\displaystyle v}$. Going from the second line to the third uses the compatibility of local and global norms. The norm is in ${\displaystyle \mathbb {Q} }$, so it remains to prove the product formula over ${\displaystyle \mathbb {Q} }$. Write

${\displaystyle a=\pm \prod _{p<\infty }p^{v_{p}},}$

where ${\displaystyle v_{p}\in \mathbb {Z} }$ is ${\displaystyle 0}$ for almost all ${\displaystyle p}$. Then

${\displaystyle {\begin{aligned}|a|&=\left(\prod _{p<\infty }|a|_{p}\right)\cdot |a|_{\infty }\\&=\left(\prod _{p<\infty }p^{-v_{p}}\right)\cdot \left(\prod _{p<\infty }p^{v_{p}}\right)\\&=1.\end{aligned}}}$

The following approximation lemma is used in the proof of compactness.

Lemma. There exists a constant ${\displaystyle C}$, depending only on ${\displaystyle K}$, such that for every ${\displaystyle \alpha =(\alpha _{v})_{v}\in \mathbb {A} _{K}}$ satisfying

${\displaystyle \prod _{v}|\alpha _{v}|_{v}>C,}$

there exists ${\displaystyle \beta \in K^{\times }}$ such that

${\displaystyle |\beta |_{v}\leq |\alpha _{v}|_{v}}$

for all ${\displaystyle v}$.¹¹

Corollary. Let ${\displaystyle v_{0}}$ be a place of ${\displaystyle K}$ and let ${\displaystyle \delta _{v}>0}$ be given for all ${\displaystyle v\neq v_{0}}$, with the property that ${\displaystyle \delta _{v}=1}$ for almost all ${\displaystyle v}$. Then there exists ${\displaystyle \beta \in K^{\times }}$ such that

${\displaystyle |\beta |_{v}\leq \delta _{v}}$

for all ${\displaystyle v\neq v_{0}}$.

Proof.

Let ${\displaystyle C}$ be the constant from the lemma. Let ${\displaystyle \pi _{v}}$ be a uniformizing element of ${\displaystyle {\mathcal {O}}_{v}}$. Define the adele ${\displaystyle \alpha =(\alpha _{v})_{v}}$ by ${\displaystyle \alpha _{v}:=\pi _{v}^{k_{v}}}$, with ${\displaystyle k_{v}\in \mathbb {Z} }$ minimal so that

${\displaystyle |\alpha _{v}|_{v}\leq \delta _{v}}$

for all ${\displaystyle v\neq v_{0}}$. Then ${\displaystyle k_{v}=0}$ for almost all ${\displaystyle v}$. Define ${\displaystyle \alpha _{v_{0}}:=\pi _{v_{0}}^{k_{v_{0}}}}$, with ${\displaystyle k_{v_{0}}\in \mathbb {Z} }$, so that

${\displaystyle \prod _{v}|\alpha _{v}|_{v}>C.}$

This works because ${\displaystyle k_{v}=0}$ for almost all ${\displaystyle v}$. By the lemma there exists ${\displaystyle \beta \in K^{\times }}$ such that

${\displaystyle |\beta |_{v}\leq |\alpha _{v}|_{v}\leq \delta _{v}}$

for all ${\displaystyle v\neq v_{0}}$.

Theorem. ${\displaystyle K^{\times }}$ is discrete and cocompact in ${\displaystyle I_{K}^{1}}$.

Proof.

Since ${\displaystyle K^{\times }}$ is discrete in ${\displaystyle I_{K}}$, it is also discrete in ${\displaystyle I_{K}^{1}}$. To prove the compactness of ${\displaystyle I_{K}^{1}/K^{\times }}$, let ${\displaystyle C}$ be the constant of the lemma and suppose ${\displaystyle \alpha \in \mathbb {A} _{K}}$ satisfies

${\displaystyle \prod _{v}|\alpha _{v}|_{v}>C.}$

Define

${\displaystyle W_{\alpha }:=\left\{\xi =(\xi _{v})_{v}\in \mathbb {A} _{K}:|\xi _{v}|_{v}\leq |\alpha _{v}|_{v}{\text{ for all }}v\right\}.}$

Clearly ${\displaystyle W_{\alpha }}$ is compact. It can be claimed that the natural projection

${\displaystyle W_{\alpha }\cap I_{K}^{1}\to I_{K}^{1}/K^{\times }}$

is surjective. Let ${\displaystyle \beta =(\beta _{v})_{v}\in I_{K}^{1}}$ be arbitrary. Then

${\displaystyle |\beta |=\prod _{v}|\beta _{v}|_{v}=1,}$

and therefore

${\displaystyle \prod _{v}|\beta _{v}^{-1}|_{v}=1.}$

It follows that

${\displaystyle \prod _{v}|\beta _{v}^{-1}\alpha _{v}|_{v}=\prod _{v}|\alpha _{v}|_{v}>C.}$

By the lemma there exists ${\displaystyle \eta \in K^{\times }}$ such that

${\displaystyle |\eta |_{v}\leq |\beta _{v}^{-1}\alpha _{v}|_{v}}$

for all ${\displaystyle v}$, and therefore ${\displaystyle \eta \beta \in W_{\alpha }}$. This proves the surjectivity of the natural projection. Since it is also continuous, compactness follows.²¹²

There is a canonical isomorphism

${\displaystyle I_{\mathbb {Q} }^{1}/\mathbb {Q} ^{\times }\cong {\widehat {\mathbb {Z} }}^{\times }.}$

Furthermore, ${\displaystyle {\widehat {\mathbb {Z} }}^{\times }\times \{1\}\subset I_{\mathbb {Q} }^{1}}$ is a set of representatives for ${\displaystyle I_{\mathbb {Q} }^{1}/\mathbb {Q} ^{\times }}$, and ${\displaystyle {\widehat {\mathbb {Z} }}^{\times }\times (0,\infty )\subset I_{\mathbb {Q} }}$ is a set of representatives for ${\displaystyle I_{\mathbb {Q} }/\mathbb {Q} ^{\times }}$.

Proof.

Consider the map

${\displaystyle {\begin{cases}\phi :{\widehat {\mathbb {Z} }}^{\times }\to I_{\mathbb {Q} }^{1}/\mathbb {Q} ^{\times },\\(a_{p})_{p}\mapsto ((a_{p})_{p},1)\mathbb {Q} ^{\times }.\end{cases}}}$

This map is well-defined, since ${\displaystyle |a_{p}|_{p}=1}$ for all ${\displaystyle p}$ and therefore

${\displaystyle \left(\prod _{p<\infty }|a_{p}|_{p}\right)\cdot 1=1.}$

Obviously ${\displaystyle \phi }$ is a continuous group homomorphism. Suppose

${\displaystyle ((a_{p})_{p},1)\mathbb {Q} ^{\times }=((b_{p})_{p},1)\mathbb {Q} ^{\times }.}$

Then there exists ${\displaystyle q\in \mathbb {Q} ^{\times }}$ such that

${\displaystyle ((a_{p})_{p},1)q=((b_{p})_{p},1).}$

By considering the infinite place it can be seen that ${\displaystyle q=1}$, which proves injectivity. To show surjectivity, let

${\displaystyle ((\beta _{p})_{p},\beta _{\infty })\mathbb {Q} ^{\times }\in I_{\mathbb {Q} }^{1}/\mathbb {Q} ^{\times }.}$

The absolute value of this element is ${\displaystyle 1}$, and therefore

${\displaystyle |\beta _{\infty }|_{\infty }={\frac {1}{\prod _{p}|\beta _{p}|_{p}}}\in \mathbb {Q} .}$

Hence ${\displaystyle \beta _{\infty }\in \mathbb {Q} }$, and there is

${\displaystyle ((\beta _{p})_{p},\beta _{\infty })\mathbb {Q} ^{\times }=\left(\left({\frac {\beta _{p}}{\beta _{\infty }}}\right)_{p},1\right)\mathbb {Q} ^{\times }.}$

Since

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

it follows that ${\displaystyle \phi }$ is surjective.

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

${\displaystyle {\begin{aligned}I_{\mathbb {Q} }&\cong I_{\mathbb {Q} }^{1}\times (0,\infty ),\\I_{\mathbb {Q} }^{1}&\cong I_{\mathbb {Q} ,\mathrm {fin} }\times \{\pm 1\}.\end{aligned}}}$

The isomorphisms are given by

${\displaystyle {\begin{cases}\psi :I_{\mathbb {Q} }\to I_{\mathbb {Q} }^{1}\times (0,\infty ),\\a=(a_{\mathrm {fin} },a_{\infty })\mapsto \left(a_{\mathrm {fin} },{\frac {a_{\infty }}{|a|}},|a|\right),\end{cases}}}$

and

${\displaystyle {\begin{cases}{\widetilde {\psi }}:I_{\mathbb {Q} ,\mathrm {fin} }\times \{\pm 1\}\to I_{\mathbb {Q} }^{1},\\(a_{\mathrm {fin} },\varepsilon )\mapsto \left(a_{\mathrm {fin} },{\frac {\varepsilon }{|a_{\mathrm {fin} }|}}\right).\end{cases}}}$

The idele norm gives the following decompositions:

${\displaystyle {\begin{aligned}I_{K}&\cong I_{K}^{1}\times M,\quad {\begin{cases}M\subset I_{K}{\text{ discrete and }}M\cong \mathbb {Z} ,&\operatorname {char} (K)>0,\\M\subset I_{K}{\text{ closed and }}M\cong \mathbb {R} _{>0},&\operatorname {char} (K)=0,\end{cases}}\\C_{K}&\cong I_{K}^{1}/K^{\times }\times N,\quad {\begin{cases}N=\mathbb {Z} ,&\operatorname {char} (K)>0,\\N=\mathbb {R} _{>0},&\operatorname {char} (K)=0.\end{cases}}\end{aligned}}}$

Proof.

First suppose ${\displaystyle \operatorname {char} (K)=p>0}$. For each place ${\displaystyle v}$ of ${\displaystyle K}$, ${\displaystyle \operatorname {char} (K_{v})=p}$, so that for all ${\displaystyle x\in K_{v}^{\times }}$, ${\displaystyle |x|_{v}}$ belongs to the subgroup of ${\displaystyle \mathbb {R} _{>0}}$ generated by ${\displaystyle p}$. Therefore, for each ${\displaystyle z\in I_{K}}$, ${\displaystyle |z|}$ is in the subgroup of ${\displaystyle \mathbb {R} _{>0}}$ generated by ${\displaystyle p}$. Thus the image of the homomorphism ${\displaystyle z\mapsto |z|}$ is a discrete subgroup of ${\displaystyle \mathbb {R} _{>0}}$. Since this group is nontrivial, it is generated by ${\displaystyle Q=p^{m}}$ for some ${\displaystyle m\in \mathbb {N} }$. Choose ${\displaystyle z_{1}\in I_{K}}$ such that ${\displaystyle |z_{1}|=Q}$. Then ${\displaystyle I_{K}}$ is the direct product of ${\displaystyle I_{K}^{1}}$ and the subgroup generated by ${\displaystyle z_{1}}$. This subgroup is discrete and isomorphic to ${\displaystyle \mathbb {Z} }$.

Now suppose ${\displaystyle \operatorname {char} (K)=0}$. For ${\displaystyle \lambda \in \mathbb {R} _{>0}}$, define

${\displaystyle z(\lambda )=(z_{v})_{v},\qquad z_{v}={\begin{cases}1,&v\nmid \infty ,\\\lambda ,&v\mid \infty .\end{cases}}}$

The map ${\displaystyle \lambda \mapsto z(\lambda )}$ is an isomorphism of ${\displaystyle \mathbb {R} _{>0}}$ onto a closed subgroup ${\displaystyle M}$ of ${\displaystyle I_{K}}$, and ${\displaystyle I_{K}\cong M\times I_{K}^{1}}$. The isomorphism is given by multiplication:

${\displaystyle {\begin{cases}\phi :M\times I_{K}^{1}\to I_{K},\\((\alpha _{v})_{v},(\beta _{v})_{v})\mapsto (\alpha _{v}\beta _{v})_{v}.\end{cases}}}$

Obviously, ${\displaystyle \phi }$ is a homomorphism. To show it is injective, let ${\displaystyle (\alpha _{v}\beta _{v})_{v}=1}$. Since ${\displaystyle \alpha _{v}=1}$ for ${\displaystyle v\nmid \infty }$, it follows that ${\displaystyle \beta _{v}=1}$ for ${\displaystyle v\nmid \infty }$. Moreover, there exists a ${\displaystyle \lambda \in \mathbb {R} _{>0}}$ such that ${\displaystyle \alpha _{v}=\lambda }$ for ${\displaystyle v\mid \infty }$. Therefore ${\displaystyle \beta _{v}=\lambda ^{-1}}$ for ${\displaystyle v\mid \infty }$. Since

${\displaystyle \prod _{v}|\beta _{v}|_{v}=1,}$

it follows that ${\displaystyle \lambda ^{n}=1}$, where ${\displaystyle n}$ is the number of archimedean places of ${\displaystyle K}$. Consequently ${\displaystyle \lambda =1}$, and therefore ${\displaystyle \phi }$ is injective.

To show surjectivity, let ${\displaystyle \gamma =(\gamma _{v})_{v}\in I_{K}}$. Define ${\displaystyle \lambda :=|\gamma |^{1/n}}$, and define ${\displaystyle \alpha _{v}=1}$ for ${\displaystyle v\nmid \infty }$ and ${\displaystyle \alpha _{v}=\lambda }$ for ${\displaystyle v\mid \infty }$. Let

${\displaystyle \beta ={\frac {\gamma }{\alpha }}.}$

Then

${\displaystyle |\beta |={\frac {|\gamma |}{|\alpha |}}={\frac {\lambda ^{n}}{\lambda ^{n}}}=1.}$

Therefore ${\displaystyle \phi }$ is surjective. The statements for ${\displaystyle C_{K}}$ follow similarly.

Let ${\displaystyle K}$ be a number field. There exists a finite set of places ${\displaystyle S}$ such that

${\displaystyle I_{K}=\left(I_{K,S}\times \prod _{v\notin S}{\mathcal {O}}_{v}^{\times }\right)K^{\times }=\left(\prod _{v\in S}K_{v}^{\times }\times \prod _{v\notin S}{\mathcal {O}}_{v}^{\times }\right)K^{\times }.}$

Proof.

The class number of a number field is finite, so let ${\displaystyle {\mathfrak {a}}_{1},\ldots ,{\mathfrak {a}}_{h}}$ be ideals representing the classes in ${\displaystyle \operatorname {Cl} _{K}}$. These ideals are generated by a finite number of prime ideals ${\displaystyle {\mathfrak {p}}_{1},\ldots ,{\mathfrak {p}}_{n}}$. Let ${\displaystyle S}$ be a finite set of places containing the archimedean places and the finite places corresponding to ${\displaystyle {\mathfrak {p}}_{1},\ldots ,{\mathfrak {p}}_{n}}$. Consider the isomorphism

${\displaystyle I_{K}/\left(\prod _{v<\infty }{\mathcal {O}}_{v}^{\times }\times \prod _{v\mid \infty }K_{v}^{\times }\right)\cong J_{K},}$

induced by

${\displaystyle (\alpha _{v})_{v}\mapsto \prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha _{v})}.}$

At infinite places the statement is immediate, so it remains to prove the statement for finite places. The inclusion ${\displaystyle \supset }$ is obvious. Let ${\displaystyle \alpha \in I_{K,\mathrm {fin} }}$. The corresponding ideal

${\displaystyle (\alpha )=\prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha _{v})}}$

belongs to a class ${\displaystyle {\mathfrak {a}}_{i}K^{\times }}$, meaning

${\displaystyle (\alpha )={\mathfrak {a}}_{i}(a)}$

for a principal ideal ${\displaystyle (a)}$. The idele ${\displaystyle \alpha '=\alpha a^{-1}}$ maps to the ideal ${\displaystyle {\mathfrak {a}}_{i}}$ under the map ${\displaystyle I_{K,\mathrm {fin} }\to J_{K}}$. That means

${\displaystyle {\mathfrak {a}}_{i}=\prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha '_{v})}.}$

Since the prime ideals in ${\displaystyle {\mathfrak {a}}_{i}}$ are in ${\displaystyle S}$, it follows that ${\displaystyle v(\alpha '_{v})=0}$ for all ${\displaystyle v\notin S}$. Thus ${\displaystyle \alpha '_{v}\in {\mathcal {O}}_{v}^{\times }}$ for all ${\displaystyle v\notin S}$. It follows that ${\displaystyle \alpha '=\alpha a^{-1}\in I_{K,S}}$, and therefore ${\displaystyle \alpha \in I_{K,S}K^{\times }}$.

The construction also extends to finite-dimensional algebras over ${\displaystyle K}$. Let ${\displaystyle A}$ be a finite-dimensional algebra over ${\displaystyle K}$. Since ${\displaystyle \mathbb {A} _{A}^{\times }}$ is not a topological group with the subspace topology in general, equip ${\displaystyle \mathbb {A} _{A}^{\times }}$ with the topology similar to ${\displaystyle I_{K}}$ above and call ${\displaystyle \mathbb {A} _{A}^{\times }}$ the idele group of ${\displaystyle A}$. The elements of the idele group are called ideles of ${\displaystyle A}$.²

Let ${\displaystyle \alpha }$ be a finite subset of ${\displaystyle A}$ containing a basis of ${\displaystyle A}$ over ${\displaystyle K}$. For each finite place ${\displaystyle v}$ of ${\displaystyle K}$, let ${\displaystyle \alpha _{v}}$ be the ${\displaystyle {\mathcal {O}}_{v}}$-module generated by ${\displaystyle \alpha }$ in ${\displaystyle A_{v}}$. There exists a finite set of places ${\displaystyle P_{0}}$ containing the archimedean places such that for all ${\displaystyle v\notin P_{0}}$, ${\displaystyle \alpha _{v}}$ is a compact subring of ${\displaystyle A_{v}}$. For each ${\displaystyle v}$, ${\displaystyle A_{v}^{\times }}$ is an open subset of ${\displaystyle A_{v}}$ and the map ${\displaystyle x\mapsto x^{-1}}$ is continuous on ${\displaystyle A_{v}^{\times }}$. As a consequence, ${\displaystyle x\mapsto (x,x^{-1})}$ maps ${\displaystyle A_{v}^{\times }}$ homeomorphically onto its image in ${\displaystyle A_{v}\times A_{v}}$. For each ${\displaystyle v\notin P_{0}}$, the group ${\displaystyle \alpha _{v}^{\times }}$ is an open and compact subgroup of ${\displaystyle A_{v}^{\times }}$.

Let ${\displaystyle P\supset P_{\infty }}$ be a finite set of places. Then

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

is an open subgroup of ${\displaystyle \mathbb {A} _{A}^{\times }}$, and ${\displaystyle \mathbb {A} _{A}^{\times }}$ is the union of all ${\displaystyle \mathbb {A} _{A}(P,\alpha )^{\times }}$. In the special case ${\displaystyle A=K}$, for each finite set of places ${\displaystyle P\supset P_{\infty }}$,

${\displaystyle \mathbb {A} _{K}(P)^{\times }=\prod _{v\in P}K_{v}^{\times }\times \prod _{v\notin P}{\mathcal {O}}_{v}^{\times }}$

is an open subgroup of ${\displaystyle \mathbb {A} _{K}^{\times }=I_{K}}$. Furthermore, ${\displaystyle I_{K}}$ is the union of all ${\displaystyle \mathbb {A} _{K}(P)^{\times }}$.

References

References

1. Neukirch 1999, Ch. VI, §1.
2. Weil 1995, Ch. IV.
3. Neukirch 1999, Ch. VI.
4. Tate 1967.
5. Ramakrishnan & Valenza 1999.
6. Cassels & Fröhlich 1967.
7. Weil 1995, Ch. VII.
8. Weil 1995.
9. Bump 1997.
10. Cassels & Fröhlich 1967, p. 69.
11. Cassels & Fröhlich 1967, p. 66.
12. Cassels & Fröhlich 1967, p. 70.

• Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, vol. 322, translated by Schappacher, Norbert, Springer, ISBN 978-3-540-65399-8.
• Weil, André (1995), Basic Number Theory, Classics in Mathematics, Springer, ISBN 978-3-540-58655-5.
• Cassels, J. W. S.; Fröhlich, Albrecht, eds. (1967), Algebraic Number Theory, London: Academic Press.
• Tate, John (1967), "Fourier analysis in number fields, and Hecke's zeta-functions", in Cassels, J. W. S.; Fröhlich, Albrecht (eds.), Algebraic Number Theory, London: Academic Press, pp. 305–347.
• Ramakrishnan, Dinakar; Valenza, Robert J. (1999), Fourier Analysis on Number Fields, Graduate Texts in Mathematics, vol. 186, Springer, ISBN 978-0-387-98436-0.
• Bump, Daniel (1997), Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics, vol. 55, Cambridge University Press, ISBN 978-0-521-65818-8.