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  "body": "\n## Statement\n\nLet $L/K$ be a finite [[Galois extension|0QMZAQ6QQ4]] of [[algebraic number fields|MWBDFS2KJ5]] with [[abelian|082GHNPJ53]] Galois group $G$.\nLet $\\mathfrak{m}$ be a modulus of $K$, as in [[class field theory|AA943HVK6Z]].\nAssume that $\\mathfrak{m}$ is divisible by every finite prime and every real place of $K$ that [[ramifies|H5RC67SD24]] in $L$.\nLet $I_K^{\\mathfrak{m}}$ be the group of [[fractional ideals|YB4JTB4WJN]] of $K$ coprime to $\\mathfrak{m}$, and let $P_K^{\\mathfrak{m}}$ be its subgroup of principal ideals $(\\alpha)$ with $\\alpha \\equiv 1 \\bmod^\\times \\mathfrak{m}$.\nThe *Artin map* $\\psi_{L/K} \\colon I_K^{\\mathfrak{m}} \\to G$ is the homomorphism that sends each prime ideal $\\mathfrak{p}$ coprime to $\\mathfrak{m}$ to its [[Frobenius element|H5145REFMW]] $\\mathrm{Frob}_{\\mathfrak{p}}$.\nThen:\n\n1. $\\psi_{L/K}$ is surjective.\n2. Assume that the finite part of $\\mathfrak{m}$ is divisible by high enough powers of the ramified primes.\nThen the kernel of $\\psi_{L/K}$ is $P_K^{\\mathfrak{m}} \\, N_{L/K}(I_L^{\\mathfrak{m}})$, where $N_{L/K}$ is the ideal norm.\nHence\n$$I_K^{\\mathfrak{m}} \\big/ P_K^{\\mathfrak{m}} N_{L/K}(I_L^{\\mathfrak{m}}) \\cong \\Gal(L/K).$$\n\n## Remarks\n\nThe smallest modulus for which part 2 holds is the conductor of $L/K$.\nIn the language of [[ideles|9GHD9MXNYV]], there is a surjective homomorphism from the idele class group of $K$ onto $G$.\nIts kernel is the image of the norm from the idele class group of $L$.\nArtin proved the law using Chebotarev's density theorem and a reduction to cyclotomic extensions.\n\nExample: $K = \\Q$, $L = \\Q(\\zeta_n)$ a [[cyclotomic field|NN77VYSGXR]], $\\mathfrak{m} = n\\infty$.\nFor a prime $p \\nmid n$, $\\psi(p\\Z)$ is the automorphism $\\zeta_n \\mapsto \\zeta_n^p$.\nSo $\\psi$ sends the ideal generated by a positive integer $a$ coprime to $n$ to $\\zeta_n \\mapsto \\zeta_n^a$.\nIts kernel is the set of ideals $(a/b)$ with $a, b \u003e 0$ coprime to $n$ and $a \\equiv b \\pmod n$, which is $P_{\\Q}^{\\mathfrak{m}}$.\nThe theorem becomes $\\Gal(\\Q(\\zeta_n)/\\Q) \\cong (\\Z/n\\Z)^\\times$.\n\n## Explanations\n\n### L1 human\n\nFor an abelian extension, the Frobenius element of an unramified prime records how that prime splits.\nArtin reciprocity says that the Frobenius element depends only on the class of the prime in a suitable generalised ideal class group of the base field.\nSo the splitting of primes in abelian extensions follows congruence rules.\n\n### L2 human\n\nThe law contains [[quadratic reciprocity|085MJS6907]] as a special case.\nLet $q$ be an odd prime, $q^* = (-1)^{(q-1)/2} q$ and $L = \\Q(\\sqrt{q^*})$.\nThe Frobenius of an odd prime $p \\ne q$ in $L$ is given by the Legendre symbol $(q^*/p)$.\nSince $L \\subset \\Q(\\zeta_q)$, it also depends only on $p$ modulo $q$.\nIt also contains the higher power reciprocity laws.\nFor $\\mathfrak{m} = 1$ and the [[Hilbert class field|WWCJ7CR2R3]] it gives an isomorphism between the class group of $K$ and the Galois group.\n",
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