Artin reciprocity law

Remarks

The smallest modulus for which part 2 holds is the conductor of $L/K$. In the language of ideles, 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, $\mathfrak{m} = n\infty$. 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 $\Gal(\Q(\zeta_n)/\Q) \cong (\Z/n\Z)^\times$.

Going deeper

For a student

The law contains quadratic reciprocity 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 it gives an isomorphism between the class group of $K$ and the Galois group.