Class Field Theory - Core Definitions

Class field theory is the culmination of algebraic number theory, describing all abelian extensions of number fields via ideal-theoretic data.

DefinitionAbelian Extension

An extension $L/K$ of number fields is abelian if it is Galois with abelian Galois group $\text{Gal}(L/K)$ .

The maximal abelian extension $K^{ab}$ is the compositum of all finite abelian extensions of $K$ . Class field theory provides an explicit description of $\text{Gal}(K^{ab}/K)$ in terms of $K$ alone.

ExampleAbelian Extensions of $\mathbb{Q}$

Cyclotomic fields: $\mathbb{Q}(\zeta_n)$ are abelian over $\mathbb{Q}$ with $\text{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}) \cong (\mathbb{Z}/n\mathbb{Z})^*$
Real cyclotomic: $\mathbb{Q}(\zeta_n)^+ = \mathbb{Q}(\zeta_n + \zeta_n^{-1})$ is maximal real subfield
Kronecker-Weber: Every abelian extension of $\mathbb{Q}$ is contained in some $\mathbb{Q}(\zeta_n)$

For $\mathbb{Q}(\sqrt{2})$ : Galois group $\mathbb{Z}/2\mathbb{Z}$ is abelian, and this equals $\mathbb{Q}(\zeta_8)^+$ .

DefinitionRay Class Group

For an ideal $\mathfrak{m}$ of $\mathcal{O}_K$ (the modulus or conductor), the ray class group is: $\text{Cl}_\mathfrak{m}(K) = I_\mathfrak{m} / P_\mathfrak{m}$

where:

$I_\mathfrak{m}$ = group of fractional ideals coprime to $\mathfrak{m}$
$P_\mathfrak{m}$ = subgroup of principal ideals $(\alpha)$ with $\alpha \equiv 1 \pmod{\mathfrak{m}}$ (and positive at specified real places)

This generalizes the ideal class group (which corresponds to $\mathfrak{m} = (1)$ ).

DefinitionArtin Map

The Artin map (or Artin symbol) is a homomorphism: $\phi_{L/K}: I_\mathfrak{m} \to \text{Gal}(L/K)$

sending each prime ideal $\mathfrak{p}$ coprime to the conductor to the Frobenius element $\text{Frob}_\mathfrak{p} \in \text{Gal}(L/K)$ .

Extends multiplicatively to all ideals and descends to ray class group, giving: $\Phi_{L/K}: \text{Cl}_\mathfrak{m}(K) \to \text{Gal}(L/K)$

TheoremArtin Reciprocity Law (Main Theorem of CFT)

For every abelian extension $L/K$ , there exists a modulus $\mathfrak{m}$ (the conductor of $L/K$ ) such that the Artin map induces an isomorphism: $\Phi_{L/K}: \text{Cl}_\mathfrak{m}(K) / N_{L/K}(\text{Cl}_\mathfrak{m}(L)) \xrightarrow{\sim} \text{Gal}(L/K)$

This establishes a bijection between abelian extensions and generalized ideal class groups, completely describing abelian extensions arithmetically.

Remark

Class field theory answers: "What are all abelian extensions of $K$ ?" via ideal-theoretic data. Unlike Galois theory (which describes extensions via roots of polynomials), CFT describes extensions via congruence conditions on ideals, providing an arithmetic classification.

The theory splits into local (for $\mathbb{Q}_p$ , much simpler) and global (for number fields, harder). The local theory was solved first and guided the global theory.