---
tag: BEKCEYHCHX
kind: concept
name: Class field theory
msc: [11R37]
sources:
  - url: https://en.wikipedia.org/wiki/Class_field_theory
same_as:
  wikidata: Q1744580
  wikipedia: Class field theory
---

## Summary

Class field theory describes the abelian extensions of a number field $K$ in terms of arithmetic data of $K$ itself.
The finite abelian extensions of $K$ correspond to subgroups of generalised ideal class groups.
Equivalently, they correspond to open subgroups of finite index of the [[idele class group|9GHD9MXNYV]].
The [[Artin reciprocity law|7938EKMMVR]] gives an isomorphism between the Galois group of such an extension and the corresponding quotient, through [[Frobenius elements|H5145REFMW]].
The [[Kronecker–Weber theorem|AW00HPFB54]] and the [[Hilbert class field|WWCJ7CR2R3]] are special cases, and [[local class field theory|SHEN158T3M]] treats the completions.

## Explanations

### L1 human

Over $\Q$, whether a prime $p$ splits in $\Q(i)$ depends only on $p$ modulo $4$.
Class field theory says this is typical.
In any extension of a number field with commutative Galois group, the way a prime splits depends only on a congruence condition on the prime.
Conversely, every such congruence condition comes from an abelian extension.
So the abelian extensions of a field are classified by data that can be read off inside the field.