---
tag: 7938EKMMVR
kind: statement
role: theorem
name: Artin reciprocity law
msc: [11R37]
uses: [MWBDFS2KJ5, 0QMZAQ6QQ4, 082GHNPJ53, H5145REFMW, YB4JTB4WJN, H5RC67SD24, BEKCEYHCHX, AA943HVK6Z, NN77VYSGXR, 085MJS6907, WWCJ7CR2R3, 9GHD9MXNYV]
about: [BEKCEYHCHX, H5145REFMW]
sources:
  - url: https://en.wikipedia.org/wiki/Artin_reciprocity
same_as:
  wikidata: Q713069
  wikipedia: Artin reciprocity
---

## Statement

Let $L/K$ be a finite [[Galois extension|0QMZAQ6QQ4]] of [[algebraic number fields|MWBDFS2KJ5]] with [[abelian|082GHNPJ53]] Galois group $G$.
Let $\mathfrak{m}$ be a modulus of $K$, as in [[class field theory|AA943HVK6Z]].
Assume that $\mathfrak{m}$ is divisible by every finite prime and every real place of $K$ that [[ramifies|H5RC67SD24]] in $L$.
Let $I_K^{\mathfrak{m}}$ be the group of [[fractional ideals|YB4JTB4WJN]] of $K$ coprime to $\mathfrak{m}$, and let $P_K^{\mathfrak{m}}$ be its subgroup of principal ideals $(\alpha)$ with $\alpha \equiv 1 \bmod^\times \mathfrak{m}$.
The *Artin map* $\psi_{L/K} \colon I_K^{\mathfrak{m}} \to G$ is the homomorphism that sends each prime ideal $\mathfrak{p}$ coprime to $\mathfrak{m}$ to its [[Frobenius element|H5145REFMW]] $\mathrm{Frob}_{\mathfrak{p}}$.
Then:

1. $\psi_{L/K}$ is surjective.
2. Assume that the finite part of $\mathfrak{m}$ is divisible by high enough powers of the ramified primes.
Then the kernel of $\psi_{L/K}$ is $P_K^{\mathfrak{m}} \, N_{L/K}(I_L^{\mathfrak{m}})$, where $N_{L/K}$ is the ideal norm.
Hence
$$I_K^{\mathfrak{m}} \big/ P_K^{\mathfrak{m}} N_{L/K}(I_L^{\mathfrak{m}}) \cong \Gal(L/K).$$

## Remarks

The smallest modulus for which part 2 holds is the conductor of $L/K$.
In the language of [[ideles|9GHD9MXNYV]], there is a surjective homomorphism from the idele class group of $K$ onto $G$.
Its kernel is the image of the norm from the idele class group of $L$.
Artin proved the law using Chebotarev's density theorem and a reduction to cyclotomic extensions.

Example: $K = \Q$, $L = \Q(\zeta_n)$ a [[cyclotomic field|NN77VYSGXR]], $\mathfrak{m} = n\infty$.
For a prime $p \nmid n$, $\psi(p\Z)$ is the automorphism $\zeta_n \mapsto \zeta_n^p$.
So $\psi$ sends the ideal generated by a positive integer $a$ coprime to $n$ to $\zeta_n \mapsto \zeta_n^a$.
Its kernel is the set of ideals $(a/b)$ with $a, b > 0$ coprime to $n$ and $a \equiv b \pmod n$, which is $P_{\Q}^{\mathfrak{m}}$.
The theorem becomes $\Gal(\Q(\zeta_n)/\Q) \cong (\Z/n\Z)^\times$.

## Explanations

### L1 human

For an abelian extension, the Frobenius element of an unramified prime records how that prime splits.
Artin reciprocity says that the Frobenius element depends only on the class of the prime in a suitable generalised ideal class group of the base field.
So the splitting of primes in abelian extensions follows congruence rules.

### L2 human

The law contains [[quadratic reciprocity|085MJS6907]] as a special case.
Let $q$ be an odd prime, $q^* = (-1)^{(q-1)/2} q$ and $L = \Q(\sqrt{q^*})$.
The Frobenius of an odd prime $p \ne q$ in $L$ is given by the Legendre symbol $(q^*/p)$.
Since $L \subset \Q(\zeta_q)$, it also depends only on $p$ modulo $q$.
It also contains the higher power reciprocity laws.
For $\mathfrak{m} = 1$ and the [[Hilbert class field|WWCJ7CR2R3]] it gives an isomorphism between the class group of $K$ and the Galois group.