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  "body": "\n## Summary\n\nClass field theory describes the abelian extensions of a number field $K$ in terms of arithmetic data of $K$ itself.\nThe finite abelian extensions of $K$ correspond to subgroups of generalised ideal class groups.\nEquivalently, they correspond to open subgroups of finite index of the [[idele class group|9GHD9MXNYV]].\nThe [[Artin reciprocity law|7938EKMMVR]] gives an isomorphism between the Galois group of such an extension and the corresponding quotient, through [[Frobenius elements|H5145REFMW]].\nThe [[Kronecker–Weber theorem|AW00HPFB54]] and the [[Hilbert class field|WWCJ7CR2R3]] are special cases, and [[local class field theory|SHEN158T3M]] treats the completions.\n\n## Explanations\n\n### L1 human\n\nOver $\\Q$, whether a prime $p$ splits in $\\Q(i)$ depends only on $p$ modulo $4$.\nClass field theory says this is typical.\nIn any extension of a number field with commutative Galois group, the way a prime splits depends only on a congruence condition on the prime.\nConversely, every such congruence condition comes from an abelian extension.\nSo the abelian extensions of a field are classified by data that can be read off inside the field.\n",
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