Adele ring ( $A_{K}$ )

For a number field

K

, its adele ring consists of tuples

(x_{v})

in the completions

K_{v}

such that

x_{v} \in O_{v}

at all but finitely many finite places. This is the restricted product

A_{K} = \prod_{v}^{'} K_{v}

, with componentwise operations and the restricted-product topology.

For

K = Q

, one has a real component and one P-adic number component for each prime. The diagonal image of a rational number belongs to the ring because only finitely many primes divide its denominator. The construction combines archimedean and congruence information in one locally compact space, enabling global harmonic analysis and formulations of class field theory.

Restricted product ( $\prod_{i}^{'} (X_{i}, U_{i})$ )

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Given spaces

X_{i}

with distinguished open subsets

U_{i}

, their restricted product consists of tuples

(x_{i})

for which

x_{i} \in U_{i}

at all but finitely many indices.

Idele group ( $I_{K}$ )

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The idele group of a number field

K

is the restricted product

I_{K} = \prod_{v}^{'} K_{v}^{\times}

with respect to

O_{v}^{\times}

at finite places. An idele is nonzero in every component and is a unit at all but finitely many finite places. Multiplication is componentwise.

This is a stronger condition than merely having an adele with no zero components: the componentwise inverse must also belong to the adele ring. For example, over

Q

, choosing the component

p

at every finite prime gives an adele but not an idele, because infinitely many components are nonunits. Quotienting by diagonal

K^{\times}

gives the idèle class group, central in class field theory.

Principal idele ( $(a)_{v}, a \in K^{\times}$ )

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An idele obtained by mapping the same nonzero element of a number field into every completion. Only finitely many local valuations are nonzero, so this diagonal tuple belongs to the idele group.

Discreteness of principal ideles

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The diagonal multiplicative group of a number field is discrete in its idele group. Require local units at every finite place and an archimedean neighbourhood of one. A diagonal element

x

meeting these restrictions has integral

x - 1

. If

x \neq = 1

, its nonzero integer field norm cannot have absolute value less than one, while sufficiently small archimedean differences would force exactly that.

Idele-to-ideal homomorphism ( $(x_{v})_{v} \mapsto \prod_{v < \infty} p_{v}^{ord_{v} (x_{v})}$ )

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The map from an idele to its finite-place fractional ideal is surjective, has kernel the product of the infinite multiplicative groups and finite-place unit groups, and sends diagonal elements to principal fractional ideals. It realizes the ideal class group as the corresponding quotient of the idele group.

Idelic modulus ( $∥ x ∥ = \prod_{v} ∣ x_{v} ∣_{v}$ )

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The idelic modulus is the product of normalized local moduli: ordinary real modulus, squared complex modulus and

(N p)^{- ord_{p} (x)}

at finite places. This continuous homomorphism to

R_{> 0}

has the diagonal multiplicative group in its kernel by the product formula. It differs from the idele norm, which maps between idele groups of field extensions.

Norm-one idele group ( $J_{K}^{1}$ )

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The norm-one idele group is the kernel of the idelic modulus. Its quotient by the diagonal

K^{\times}

is compact for a number field and surjects onto the ideal class group.

An idele is an element of the idele group: a tuple of nonzero elements of the completion of a valued field at all places, with finite-place components in the local unit groups except at finitely many places.

Restricted product topology on the idele group

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The idele topology is the restricted product topology with respect to

O_{v}^{\times}

. It is also the subspace topology induced by

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

from

A_{K} \times A_{K}

Idèle class group ( $C_{K}$ )

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The idele class group is

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

, where

K^{\times}

is embedded diagonally.

Idelic congruence subgroup ( $C_{K} (m)$ )

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For a modulus of a number field

m

, form a subgroup

U (m)

of the idele group as follows. At a finite prime dividing

m_{0}

with exponent

n

, use

1 + p^{n} O_{K_{p}}

; at every other finite prime, use

O_{K_{p}}^{\times}

. At a real place selected by

m_{\infty}

, use

R_{> 0}

, and at every other infinite place use the entire multiplicative completion.

The image of

U (m)

in the idèle class group

C_{K}

is the idelic congruence subgroup

C_{K} (m)

. Its quotient is the ray class group. The unit requirement outside the modulus is essential: allowing arbitrary valuations at those primes would discard ideal-class information. Multiplication by a principal idele accounts for changing a global generator while preserving its ideal class.

Norm group of an abelian extension ( $N_{L / K} C_{L}$ )

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For a finite abelian extension

L / K

, the norm group is the image of the idele-class norm

C_{L} \to C_{K}

. Class field theory gives

C_{K} / N_{L / K} C_{L} ≅ Gal (L / K)

Idele norm ( $N_{L / K} : I_{L} \to I_{K}$ )

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The

v

-component of the idele norm is

\prod_{w ∣ v} N_{L_{w} / K_{v}} (x_{w})

. It descends to an idèle class group norm

C_{L} \to C_{K}

Nonabelian norm-residue kernel theorem

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For a finite Galois extension

L / K

with group

G

, the kernel of the global reciprocity map

C_{K} \to G^{ab}

N_{L / K} C_{L}

. It follows that this norm group equals the norm group from the maximal abelian subextension

L^{[G, G]} / K

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The "Adele ring" refers to a specific type of ring associated with the singer Adele, particularly her engagement ring. Adele's engagement ring is notable for its intricate design and has garnered attention in the media due to the artist's high profile. The ring is often described as a large diamond set in a unique design, highlighting Adele's style and taste. Additionally, there may be references to "Adele rings" in popular culture or jewelry trends inspired by her aesthetic.

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