Artin reciprocity law

The Artin map sends an ideal prime to the ramified primes to the product of its Frobenius elements. For a suitable modulus, its kernel is the norm or congruence subgroup associated with the abelian extension.

For a finite abelian extension of number fields, local reciprocity maps into the local decomposition groups combine to a continuous map on the idele group: outside finitely many places they kill units, and an idele has nonunit components only finitely often. The Artin reciprocity law is equivalent to the compatibility that this product kills every principal idele. Conversely, restrict a global continuous Artin homomorphism to each single-place embedding of $K_v^{\ast }$. These maps reconstruct the global map, by density of finite-component ideles and triviality on a basic open subgroup of units. Merely having unrelated local maps omits the essential principal-idele compatibility.

For an abelian extension $L/K$ of number fields, restriction to an intermediate field takes each Artin symbol to that of the smaller extension. For a finite base extension $E/K$, restriction identifies $\mathrm{Gal}(LE/E)$ with $\mathrm{Gal}(L/(L\cap E))$. At an unramified prime $\mathfrak{q}$ of $E$ above $\mathfrak{p}$, its residue-field Frobenius restricts to the $f(\mathfrak{q}/\mathfrak{p})$th power of the Frobenius automorphism at $\mathfrak{p}$. Since $N_{E/K}\mathfrak{q}={\mathfrak{p}}^{f(\mathfrak{q}/\mathfrak{p})}$, the group homomorphism property proves the displayed identity for fractional ideals supported over unramified primes.