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  "body": "\n## Statement\n\nLet $K$ be a [[global field|634XKY8RZ0]] and let $A$ be a [[central simple algebra|5HRKJBAY2V]] over $K$.\nFor each place $v$ of $K$, let $K_v$ be the completion of $K$ at $v$, a [[local field|J85FZEQG38]] or $\\R$ or $\\C$.\nIf $K_v$ [[splits|NVD7R0T835]] $A$, that is, $A \\otimes_K K_v$ is a matrix algebra over $K_v$, for every place $v$, then $A$ is a matrix algebra over $K$.\nEquivalently, the natural map of [[Brauer groups|BEW48TA9WZ]] $\\mathrm{Br}(K) \\to \\bigoplus_v \\mathrm{Br}(K_v)$ is injective.\n\n## Remarks\n\nThe proof uses class field theory: one reduces to cyclic algebras and then uses the Hasse norm theorem.\nThat theorem says that in a cyclic extension of global fields, an element that is a local norm everywhere is a global norm.\nExample: the quaternion algebra $\\left(\\frac{-1, -1}{\\Q}\\right)$ splits over $\\Q_p$ for every odd prime $p$, but not over $\\Q_2$ or over $\\R$.\nSo it is not split over $\\Q$, in agreement with the theorem.\nClass field theory gives more: the sequence $0 \\to \\mathrm{Br}(K) \\to \\bigoplus_v \\mathrm{Br}(K_v) \\to \\Q/\\Z \\to 0$, with the sum of local invariants as the last map, is exact.\nBecause of this sum rule it is enough to check all places but one, and a quaternion algebra over $K$ is ramified at an even number of places.\n\n## Explanations\n\n### L1 human\n\nIs a given central simple algebra over the rational numbers just a matrix algebra?\nTo decide, it is enough to check it over the real numbers and over each field of $p$-adic numbers.\nThese local questions are much easier.\nThe theorem is a local-global principle, like the Hasse–Minkowski theorem for quadratic forms.\n\n### L2 human\n\nThe theorem is a cornerstone of the arithmetic of central simple algebras over number fields.\nWith the exact sequence above it gives the classification of central simple algebras over a global field by their local invariants.\nConsequences are that every central simple algebra over a global field is cyclic and that its period equals its index.\nFor quaternion algebras it reduces to the Hasse–Minkowski theorem for the ternary quadratic form $a x^2 + b y^2 - a b z^2$.\n",
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