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  "body": "\n## Statement\n\nLet $K$ be a field, let $\\ell$ be a [[prime number|0SN0MAEKYX]] different from the [[characteristic|1ZB13N78XW]] of $K$, and let $n \\ge 0$.\nLet $K^s$ be a [[separable closure|JJ91A1V8NQ]] of $K$, and let $\\mu_\\ell \\subseteq K^s$ be the group of $\\ell$-th roots of unity.\nLet $\\mu_\\ell^{\\otimes n}$ be its $n$-fold tensor power over $\\Z$, with $\\mu_\\ell^{\\otimes 0} = \\Z/\\ell\\Z$.\nLet $\\delta \\colon K^\\times \\to H^1(K, \\mu_\\ell)$ be the Kummer map.\nThen the norm residue map\n$$K_n^M(K)/\\ell \\longrightarrow H^n(K, \\mu_\\ell^{\\otimes n}), \\qquad \\set{a_1, \\ldots, a_n} \\longmapsto \\delta(a_1) \\cup \\cdots \\cup \\delta(a_n),$$\nfrom [[Milnor K-theory|2YXDQ7Q445]] modulo $\\ell$ to [[Galois cohomology|EESNTXFWHZ]], is well defined and is an isomorphism of groups.\n\n## Remarks\n\nThe map is well defined because the cup product $\\delta(a) \\cup \\delta(1 - a)$ vanishes for $a \\ne 0, 1$.\nFor $n = 0$ the statement is trivial, and for $n = 1$ it is Kummer theory with [[Hilbert's Theorem 90|YFF8B5A0QG]]: $K^\\times/(K^\\times)^\\ell \\cong H^1(K, \\mu_\\ell)$.\nFor $n = 2$ it is the Merkurjev–Suslin theorem, which identifies $K_2(K)/\\ell$ with the $\\ell$-torsion of the [[Brauer group|BEW48TA9WZ]] when $K$ contains the $\\ell$-th roots of unity.\nThe case $\\ell = 2$ is the Milnor conjecture, proved by Voevodsky; the general case, formerly the Bloch–Kato conjecture, was completed by Voevodsky with essential contributions of Rost.\nThe proof uses motivic cohomology and is far beyond elementary methods.\nThe theorem also holds with $\\ell$ replaced by any positive integer $m$ invertible in $K$, with $\\mu_m^{\\otimes n}$ in place of $\\mu_\\ell^{\\otimes n}$.\n\n## Explanations\n\n### L1 human\n\nMilnor K-groups are defined by simple generators and relations, while Galois cohomology groups are defined through the symmetries of field extensions.\nThe theorem says that, modulo a prime $\\ell$, these two very different constructions give the same groups.\nSo questions about Galois cohomology can be answered with explicit symbols, and the other way round.\n\n### L2 human\n\nIn degree $2$, suppose $K$ contains the $\\ell$-th roots of unity.\nThen every central simple algebra of exponent $\\ell$ is Brauer equivalent to a tensor product of cyclic algebras.\nFor $\\ell = 2$ it yields, with further work, the description of the graded Witt ring of quadratic forms by Galois cohomology, also called the Milnor conjecture on quadratic forms.\n",
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