Local class field theory

Local class field theory describes the abelian extensions of a local field $K$ through its multiplicative group $K^{\times }$.

Local Artin reciprocity gives a continuous homomorphism ${\mathrm{Art}}_K:K^{\times }\to \mathrm{Gal}(K^{\mathrm{ab}}/K)$ with dense image. For every finite abelian extension $L/K$, it induces an isomorphism

The local Artin map is the reciprocity homomorphism of Local Artin reciprocity. Its normalization is fixed by choosing whether a uniformizer maps to arithmetic or geometric Frobenius.

For a finite extension of local fields $L/K$, its norm subgroup is the image of the field norm $N_{L/K}:L^{\times }\to K^{\times }$. For an abelian extension, it is the kernel of the induced local Artin map to $\mathrm{Gal}(L/K)$.