Adele ring ( $A_{K}$ )

The adele ring is the restricted product of the completions

K_{v}

with respect to the valuation rings

O_{v}

at finite places.

Table of contents
- Restricted product Adele ring
  - Restricted product topology Restricted product
- Idele group Adele ring
  - Restricted product topology on the idele group Idele group
  - Idèle class group Idele group
    - Idelic congruence subgroup Idèle class group
    - Norm group of an abelian extension Idèle class group
      - Idele norm Norm group of an abelian extension
      - Nonabelian norm-residue kernel theorem Norm group of an abelian extension

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

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.

Restricted product topology

A basic open set in a restricted product

\prod_{v}^{'} (X_{v}, U_{v})

is a product of open local sets that equals the distinguished subset

U_{v}

at all but finitely many indices.

Idele group ( $I_{K}$ )

The idele group is the restricted product

\prod_{v}^{'} K_{v}^{\times}

with respect to

O_{v}^{\times}

at finite places.

Restricted product topology on the idele group

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 ( $C_{K}$ )

The idele class group is

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

, where

K^{\times}

is embedded diagonally.

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

The ideles congruent to one modulo

m

have finite components in

1 + p^{n_{p}}

at primes in the finite modulus and positive components at selected real places. Their image in

C_{K}

is the idelic congruence subgroup.

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

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

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

For a finite Galois extension

L / K

with group

G

, the kernel of the global reciprocity map

C_{K} \to G^{ab}

is

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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