Idele groups and class field theory are powerful tools in algebraic number theory. They connect local and global properties of number fields, providing a unified framework for studying abelian extensions and ideal class groups.

These concepts build on adeles, extending their applications to field extensions and reciprocity laws. Ideles offer a refined view of number fields, enabling deeper insights into their arithmetic structure and relationships between different algebraic objects.

Idele groups from adele rings

Construction and properties of idele groups

• Define idele groups as invertible elements in the adele ring of a number field
• Construct idele groups by selecting units from each local component of the adele ring
• Equip idele group JK of number field K with topology making it a locally compact topological group
• Topology of JK refined beyond subspace topology inherited from adele ring
• Define idele class group CK as quotient of JK by image of K under diagonal embedding
• Idele class group CK exhibits compactness, crucial for class field theory applications

Canonical homomorphisms and unit ideles

• Establish canonical surjective homomorphism from JK to ideal group IK of K
• Map each idele to corresponding fractional ideal
• Identify kernel of JK to IK homomorphism as group of unit ideles
• Unit ideles consist of ideles with unit components at all finite places
• Surjective nature of homomorphism connects idele structure to ideal structure
• Kernel structure provides insight into relationship between local and global properties

Idele groups and ideal class groups

Quotient realizations and homomorphisms

• Realize ideal class group of number field K as quotient of idele class group CK
• Establish canonical surjective homomorphism from CK to ideal class group ClK
• Connect kernel of CK to ClK homomorphism with group of principal ideals
• Deduce finiteness of ideal class group from compactness of CK and discreteness of K image in JK
• Unify treatment of finite and infinite places of K through idele formulation
• Enable study of generalized ideal class groups with modulus, crucial for comprehensive class field theory

Applications to number theory

• Provide natural setting for studying L-functions and zeta functions of number fields
• Offer refined view of ideal class group through idele class group connection
• Facilitate exploration of arithmetic properties through idelic structure
• Enable investigation of class numbers and related invariants using idelic approach
• Support analysis of prime decomposition in number field extensions via idelic formulation
• Enhance understanding of Dirichlet unit theorem through idelic perspective

Class field theory with ideles

Main theorems and correspondences

• State Existence Theorem: open subgroups of finite index in CK correspond to unique abelian extensions L/K
• Assert Isomorphism Theorem: Galois group of maximal abelian extension isomorphic to profinite completion of CK
• Formulate Conductor-Discriminant Formula using ideles, relating conductor, discriminant, and local factors
• Establish one-to-one correspondence between finite abelian extensions of K and open subgroups of finite index in CK (Main Theorem)
• Describe decomposition and inertia groups of primes in abelian extensions using idelic formulation
• Generalize class field theory to global fields of positive characteristic, unifying number and function fields

Applications to field extensions

• Provide natural framework for studying Hilbert class field of number field K
• Facilitate exploration of ray class fields using idelic approach
• Enable precise description of norm groups for abelian extensions
• Support investigation of ramification in abelian extensions through idelic formulation
• Offer tools for analyzing splitting of primes in abelian extensions
• Enhance understanding of Galois groups of abelian extensions using idelic structure

Artin reciprocity law with ideles

Idelic formulation and isomorphisms

• Establish Artin reciprocity law isomorphism between CK modulo norm group of L/K and Gal(L/K)
• Define idelic Artin map as homomorphism from JK to Gal(L/K)
• Induce isomorphism in Artin reciprocity law through idelic Artin map
• Identify kernel of idelic Artin map for L/K as norms of ideles from L to K
• Unify local reciprocity laws at all places of K (finite and infinite) through idelic formulation
• Express compatibility of Artin reciprocity with restriction and inflation of Galois groups using ideles

Applications and special cases

• Provide powerful tool for studying prime splitting behavior in abelian extensions
• Interpret explicit reciprocity laws (power residue symbols) as special cases of idelic Artin reciprocity
• Enable computation of Frobenius elements in Galois groups using idelic formulation
• Facilitate study of reciprocity laws in higher local fields through idelic approach
• Support investigation of Langlands program connections using idelic Artin reciprocity
• Enhance understanding of class field theory's local-global principles via idelic formulation