Adele ring

The nobility ring is defined in mathematics in the context of class field theory and allows a particularly elegant representation of the Artin reciprocity law.

If a global field, ie either an algebraic number field or an algebraic function field over a finite field, there is the noble ring of all elements in which almost all of the components are quite (a non- negative valuation have ). Here are the set of reviews of and the completions of respect to the reviews. With a certain coarsening of the product topology (almost all components equal ) is the noble ring to a locally compact topological ring (ie, the links are continuous maps ). The group of units is the Idelgruppe. This carries the subspace topology of the product topology.

The units of the nobility ring form the Idelgruppe.

Explicit construction

Is Z, the pro- completion of the finite integers, ie the inverse limit of the rings Z / nZ:

From the Chinese Remainder Theorem the isomorphism follows

With the product of the rings of p- adic numbers.

The ring of integer Adele AZ is the product

The ring of ( rational ) Adele AQ is the tensor product

For an algebraic number field F AF is defined as

The bijection with the product of copies of AQ allows the definition of the product topology.

The invertible elements in the ring needle are referred to as Idele.

Applications

Generalizations of the Artin reciprocity law lead to the connections between automorphic representations (special representations of the general linear group) and Galois representations of ( Langlands program).

• Algebraic Number Theory