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  "body": "\n## Summary\n\nThe Brauer group of a field $K$ is the group of classes of central simple algebras over $K$ up to matrix algebras, see the [[definition|EX34J8PP61]].\nIts product is given by the tensor product.\nEach class contains exactly one central [[division ring|0EA5BTA0CX]] up to isomorphism, so the group classifies the division algebras with center $K$ and finite dimension.\nIt is isomorphic to the Galois cohomology group $H^2(K, (K^s)^\\times)$, see [[Galois cohomology|9MPCZ3EAJM]].\nIts computation for local and global fields is a main part of class field theory.\n\n## Explanations\n\n### L1 human\n\nOver the real numbers there are exactly two finite-dimensional division algebras with center $\\R$: $\\R$ itself and the quaternions.\nThe Brauer group packs this kind of information into a group.\nEach division algebra with center $K$ gives an element, and the product of two elements comes from the tensor product of the algebras.\nFor $\\R$ the group has two elements, and the quaternions square to the trivial class, because $\\mathbb{H} \\otimes_\\R \\mathbb{H}$ is the algebra of $4 \\times 4$ real matrices.\n",
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