15.2 Existence theorem and uniqueness theorem

Class Field Theory's Existence and Uniqueness Theorems are game-changers. They show that every finite abelian extension of a number field corresponds to a congruence subgroup of a ray class group. This gives us a way to describe and construct all abelian extensions.

These theorems are the backbone of Class Field Theory. They establish a one-to-one correspondence between abelian extensions and congruence subgroups, allowing us to classify and study these extensions systematically. This connection is crucial for understanding the structure of number fields.

Class field existence theorem

Statement and implications

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• Existence theorem states for every finite abelian extension K/k of number fields, a modulus m exists such that K is the class field of a congruence subgroup H of the ray class group modulo m
• Modulus m determines ramification behavior of extension K/k
• Congruence subgroup H corresponds to Galois group of K/k via class field theory
• Implies all finite abelian extensions of a number field can be described using ideal class groups and congruence conditions
• Provides method to construct all abelian extensions of a given number field
• Relates closely to Kronecker-Weber theorem for cyclotomic extensions of rational numbers

Proof techniques and applications

• Proof typically involves complex analytic methods
• Utilizes L-functions and theory of zeta functions
• Crucial for understanding structure of algebraic number fields
• Allows classification of abelian extensions based on congruence subgroups
• Enables construction of class fields for given modulus and congruence subgroup
• Facilitates study of ramification behavior in abelian extensions
• Provides foundation for more advanced topics in algebraic number theory (Langlands program, explicit class field theory)

Class field uniqueness theorem

Statement and proof outline

• Uniqueness theorem states if K1 and K2 are class fields corresponding to same congruence subgroup H of ray class group modulo m, then K1 = K2
• Proof relies on properties of Artin map and Frobenius automorphisms for unramified prime ideals
• Key step involves showing Frobenius elements generate entire Galois group of extension
• Utilizes properties of Galois theory and ideal decomposition in number fields
• Demonstrates one-to-one correspondence between congruence subgroups and abelian extensions

Consequences and applications

• Ensures correspondence between congruence subgroups and abelian extensions is one-to-one
• Defines conductor of class field K/k as minimal modulus m for which K is class field of congruence subgroup modulo m
• Allows classification of all abelian extensions of a number field using congruence subgroups of ray class groups
• Combined with existence theorem, provides complete description of abelian extensions of a number field
• Facilitates study of ramification and splitting behavior of prime ideals in abelian extensions
• Enables computation of Galois groups for abelian extensions
• Supports development of explicit class field theory and computational methods in algebraic number theory

Constructing class fields

Process and techniques

• Select modulus m and congruence subgroup H of ray class group modulo m for base field k
• Apply existence theorem to ensure class field K exists for chosen congruence subgroup H
• Utilize uniqueness theorem to confirm K is unique abelian extension corresponding to H
• Degree of class field K over k equals index of H in full ray class group modulo m
• Galois group Gal(K/k) isomorphic to quotient of ray class group modulo m by congruence subgroup H
• Construct splitting field of class equation (polynomial whose roots generate K over k)
• Use ideal theoretic methods (decomposition and inertia groups) to analyze ramification of prime ideals

Applications and examples

• Construct Hilbert class field as maximal unramified abelian extension
• Generate ray class fields for given modulus to study ramification at specific primes
• Construct cyclotomic fields as class fields over rational numbers
• Create abelian extensions with prescribed Galois groups and ramification behavior
• Analyze decomposition of prime ideals in constructed class fields
• Compute class numbers and unit groups of number fields using class field constructions
• Apply class field constructions to solve Diophantine equations and study rational points on curves

Class field theory relationships

Artin reciprocity law connections

• Artin reciprocity law establishes isomorphism between Galois group of abelian extension K/k and quotient of idele class group of k by norm group of K
• Existence and uniqueness theorems provide foundation for Artin reciprocity law
• Ensure one-to-one correspondence between abelian extensions and congruence subgroups
• Artin reciprocity law generalizes existence and uniqueness theorems to idele class groups
• Existence theorem corresponds to surjectivity of Artin map in reciprocity law
• Uniqueness theorem relates to injectivity of Artin map

Broader implications and applications

• Conductor-discriminant formula connects conductor of abelian extension to its discriminant
• Relationship provides tool for studying arithmetic of number fields
• Enables analysis of prime ideal distribution and splitting behavior
• Crucial for applications in L-functions, Langlands program, and explicit class field theory
• Supports development of advanced topics in algebraic number theory (Iwasawa theory, Stark conjectures)
• Facilitates computation of class numbers and unit groups for number fields
• Provides framework for studying non-abelian extensions and higher-dimensional class field theory