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  "body": "\n## Summary\n\nLocal class field theory describes the abelian extensions of a non-Archimedean [[local field|J85FZEQG38]] $K$ in terms of the multiplicative group $K^\\times$.\nIts main result is the local reciprocity map $K^\\times \\to \\Gal(K^{ab} / K)$, where $K^{ab}$ is the maximal abelian extension of $K$.\nFor each finite abelian extension $L / K$, it induces an isomorphism $K^\\times / N_{L/K}(L^\\times) \\cong \\Gal(L / K)$.\nThe finite abelian extensions of $K$ correspond to the open subgroups of finite index of $K^\\times$, through their norm groups.\n\n## Explanations\n\n### L1 human\n\nClass field theory says that the abelian extensions of a field are controlled by data inside the field itself.\nOver a local field such as $\\Q_p$ the data is simple: the group of nonzero elements $K^\\times$.\nEach abelian extension corresponds to a subgroup of $K^\\times$ of finite index, namely the norms from the extension.\nThe Galois group of the extension is the quotient of $K^\\times$ by this subgroup.\nFor example, take $K = \\Q_p$ and its unramified extension of degree $n$.\nIt corresponds to the elements whose [[p-adic absolute value|WGN82TCEF2]] is an integer power of $p^n$.\nIt is the local half of [[class field theory|BEKCEYHCHX]]; the global theory glues the local ones together.\n",
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