Class Field Theory — Mathematics Knowledge System

Overview

Class field theory is the crown jewel of algebraic number theory. It provides a complete classification of abelian extensions of a number field $K$ in terms of arithmetic data intrinsic to $K$ — specifically, its idèle class group. The Artin reciprocity map generalises both quadratic reciprocity and Dirichlet's theorem on primes in progressions.

Intuition

Over $\mathbb{Q}$, the Kronecker–Weber theorem says every abelian extension is contained in a cyclotomic field $\mathbb{Q}(\zeta_n)$. Class field theory generalises this: for any number field $K$, every abelian extension $L/K$ is 'controlled' by a generalised ideal class group of $K$, and primes that split completely in $L$ are characterised by congruence conditions.

Formal Definition

Definition

Let $K$ be a number field and $\mathbb{A}_K^\times$ its idèle group.

\text{Art}_K : \mathbb{A}_K^\times / K^\times \xrightarrow{\;\sim\;} \text{Gal}(K^{\text{ab}} / K)

Global Artin reciprocity map: isomorphism of the idele class group onto the Galois group of the maximal abelian extension

artin-map

\text{Frob}_\mathfrak{p} \in \text{Gal}(L/K), \quad \text{Frob}_\mathfrak{p}(x) \equiv x^{N\mathfrak{p}} \pmod{\mathfrak{P}}

Frobenius element at an unramified prime P above p

frobenius

\mathfrak{p} \text{ splits completely in } L/K \iff \text{Frob}_\mathfrak{p} = 1 \iff \mathfrak{p} \in \ker(\text{Art}_{L/K})

Splitting criterion via the Artin map

split-criterion

Notation

Notation	Meaning
$K^{\text{ab}}$	Maximal abelian extension of K
$\text{Art}_K$	Global Artin reciprocity map
$\text{Frob}_\mathfrak{p}$	Frobenius element at prime p
$I_K(\mathfrak{m})$	Group of fractional ideals coprime to modulus m
$\text{Cl}_K(\mathfrak{m})$	Ray class group modulo m

Theorems

Theorem 1: Artin Reciprocity

For an abelian extension L/K with conductor \mathfrak{m}, there is a surjective homomorphism \text{Art}_{L/K}: I_K(\mathfrak{m}) \to \text{Gal}(L/K) with kernel equal to the subgroup of principal ideals generated by elements \equiv 1 \pmod{\mathfrak{m}}.

Theorem 2: Kronecker–Weber Theorem

Every finite abelian extension of \mathbb{Q} is contained in a cyclotomic field \mathbb{Q}(\zeta_n) for some n \geq 1.

Theorem 3: Existence Theorem

For every open subgroup H of finite index in \mathbb{A}_K^\times / K^\times, there exists a unique abelian extension L/K such that H = \ker(\text{Art}_K) and \text{Gal}(L/K) \cong \mathbb{A}_K^\times / (K^\times H).

Worked Examples

$\mathbb{Q}(\sqrt{5})/\mathbb{Q}$ is abelian (Galois group $\mathbb{Z}/2\mathbb{Z}$). By Kronecker–Weber it lies in $\mathbb{Q}(\zeta_n)$ for some $n$.
The discriminant of $\mathbb{Q}(\sqrt{5})$ is 5. The conductor should divide 5. Check: $\mathbb{Q}(\zeta_5)$ has Galois group $(\mathbb{Z}/5\mathbb{Z})^\times \cong \mathbb{Z}/4\mathbb{Z}$; its unique quadratic subfield is $\mathbb{Q}(\sqrt{5^*})$ where $5^* = 5 \cdot (-1)^{(5-1)/2} = 5$.
$\mathbb{Q}(\sqrt{5}) \subset \mathbb{Q}(\zeta_5)$

Answer: $\mathbb{Q}(\sqrt{5}) \subset \mathbb{Q}(\zeta_5)$; the conductor is 5.

Practice Problems

Difficulty 8/10

List all quadratic extensions of $\mathbb{Q}$ that are contained in $\mathbb{Q}(\zeta_{12})$.

Difficulty 9/10

Which primes $p$ split completely in $\mathbb{Q}(\zeta_7)$?

Difficulty 10/10

Sketch the proof that every unramified abelian extension $L/K$ is contained in the Hilbert class field $H_K$.

Historical Background

Kronecker's Jugendtraum (youthful dream) sought to generate abelian extensions of imaginary quadratic fields via special values of the $j$-function. Hilbert's 12th problem asked for an explicit construction in general. Takagi classified all abelian extensions of number fields in 1920; Artin proved his reciprocity law in 1927, unifying earlier reciprocity theorems. Chevalley reformulated the theory using idèles in 1936, giving the modern adèlic framework.

1853
Kronecker states his Jugendtraum about abelian extensions
Leopold Kronecker
1920
Takagi classifies all abelian extensions via ray class groups
Teiji Takagi
1927
Artin proves his reciprocity law and introduces the Artin map
Emil Artin
1936
Chevalley introduces idèles and reformulates class field theory
Claude Chevalley

Summary

Class field theory classifies all abelian extensions of a number field $K$ via its idèle class group.
The Artin reciprocity map $\mathbb{A}_K^\times / K^\times \xrightarrow{\sim} \text{Gal}(K^{\text{ab}}/K)$ is the central isomorphism.
Kronecker–Weber: every abelian extension of $\mathbb{Q}$ lies in a cyclotomic field.
Primes split completely in $L/K$ iff their Frobenius is trivial, characterised by congruence conditions.
The Hilbert class field is the maximal unramified abelian extension; its degree equals $h_K$.

References

BookNeukirch, J. Class Field Theory. Springer, 1986.
BookMilne, J. Class Field Theory (lecture notes). Available at www.jmilne.org.