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  "body": "\n## Definition\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]] with ring of integers $\\mathcal{O}_K$.\nA *modulus* of $K$ is a formal product $\\mathfrak{m} = \\mathfrak{m}_0 \\mathfrak{m}_\\infty$ of a nonzero ideal $\\mathfrak{m}_0$ of $\\mathcal{O}_K$ and a set $\\mathfrak{m}_\\infty$ of real embeddings of $K$.\nLet $I_K^{\\mathfrak{m}}$ be the group of [[fractional ideals|YB4JTB4WJN]] of $K$ coprime to $\\mathfrak{m}_0$.\nLet $P_K^{\\mathfrak{m}}$ be its [[subgroup|0X1JXT47GA]] of principal ideals $(\\alpha)$ with $\\alpha \\in K^\\times$ such that $v_{\\mathfrak{p}}(\\alpha - 1) \\ge v_{\\mathfrak{p}}(\\mathfrak{m}_0)$ for every prime $\\mathfrak{p} \\mid \\mathfrak{m}_0$ and $\\sigma(\\alpha) \u003e 0$ for every $\\sigma \\in \\mathfrak{m}_\\infty$.\nA *congruence subgroup* modulo $\\mathfrak{m}$ is a subgroup $H$ with $P_K^{\\mathfrak{m}} \\subseteq H \\subseteq I_K^{\\mathfrak{m}}$; the quotient $I_K^{\\mathfrak{m}}/P_K^{\\mathfrak{m}}$ is the *ray class group* modulo $\\mathfrak{m}$.\nLet $L/K$ be a finite [[Galois extension|0QMZAQ6QQ4]].\nIt is the *class field* of $H$ if the following holds for every nonzero prime ideal $\\mathfrak{p}$ of $\\mathcal{O}_K$ coprime to $\\mathfrak{m}_0$.\nThe prime $\\mathfrak{p}$ [[splits completely|B7KNZCBFYT]] in $L$ if and only if $\\mathfrak{p} \\in H$.\nClass field theory is the theory of class fields of congruence subgroups.\n\n## Remarks\n\nThe main theorems are as follows.\nEach congruence subgroup $H$ has a class field $L$, unique inside a fixed algebraic closure.\nThis $L/K$ is [[abelian|082GHNPJ53]] and unramified at the places not dividing $\\mathfrak{m}$.\nThe [[Artin map|7938EKMMVR]] gives $\\Gal(L/K) \\cong I_K^{\\mathfrak{m}}/H$.\nEvery finite abelian extension of $K$ is the class field of some congruence subgroup.\nThe ray class group is finite, and its order is a multiple of the class number.\n\nExample: $K = \\Q$, $\\mathfrak{m} = n\\infty$ with $n \\ge 3$, and $H = P_{\\Q}^{\\mathfrak{m}}$.\nFor a prime $p \\nmid n$, the ideal $(p)$ has the unique positive generator $p$, so $(p) \\in H$ if and only if $p \\equiv 1 \\pmod n$.\nThis is exactly the condition for $p$ to split completely in the [[cyclotomic field|NN77VYSGXR]] $\\Q(\\zeta_n)$, so $\\Q(\\zeta_n)$ is the class field of $H$.\n\nNon-example: for $\\mathfrak{m} = 5$ without the infinite place, $(p) = (-p)$ lies in $P_{\\Q}^{\\mathfrak{m}}$ when $p \\equiv \\pm 1 \\pmod 5$.\nSo $\\Q(\\zeta_5)$ is not the class field of $P_{\\Q}^{\\mathfrak{m}}$: the prime $19 \\equiv -1 \\pmod 5$ gives $(19) \\in P_{\\Q}^{\\mathfrak{m}}$, but $19$ does not split completely in $\\Q(\\zeta_5)$.\nThe class field of this group is the real subfield $\\Q(\\zeta_5 + \\zeta_5^{-1}) = \\Q(\\sqrt{5})$.\n",
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