Maximal abelian extension ($K^{\mathrm{ab}}$)

The maximal abelian extension $K^{\mathrm{ab}}$ of a field $K$ is the compositum in a fixed separable closure of all finite abelian extensions of $K$.

For a non-Archimedean local field $K$, the Weil group is the inverse image of the infinite cyclic subgroup generated by Frobenius under $\mathrm{Gal}(\overline{K}/K)\to \mathrm{Gal}(\overline{k}/k)\cong \hat{\mathbb{Z}}$. It is dense in the absolute Galois group with its profinite topology.

The cyclotomic extension ${\mathbb{Q}}_p({\zeta }_{p^n})/{\mathbb{Q}}_p$ is generated by a primitive $p^n$th root of unity. It is totally ramified and has Galois group $(\mathbb{Z}/p^n\mathbb{Z}{)}^{\times }$.

The local Kronecker-Weber theorem states that every finite abelian extension of ${\mathbb{Q}}_p$ is contained in an extension obtained by adjoining roots of unity.

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 $\mathrm{Gal}(L/K)$.