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  "body": "\n## Summary\n\nA crossed product algebra is a central simple algebra built from a finite [[Galois extension|0QMZAQ6QQ4]] $L/K$.\nThe extra datum is a $2$-cocycle of its [[Galois group|0XTY86BZKY]] with values in $L^\\times$.\nIt contains $L$ as a maximal subfield, and each Galois automorphism is realized as conjugation by an invertible element.\nEvery central simple algebra is Brauer equivalent to a crossed product, which gives the isomorphism between the [[Brauer group|BEW48TA9WZ]] and a second [[Galois cohomology|9MPCZ3EAJM]] group.\nThis algebraic notion is distinct from the [[crossed product|8HG8TS46A6]] of operator algebras, although both twist a ring by a group action.\n\n## Explanations\n\n### L1 human\n\nStart with a Galois extension $L$ of $K$ and its group of symmetries $G$.\nAdd to $L$ one new symbol $u_\\sigma$ for each symmetry $\\sigma$, and declare that moving a number past $u_\\sigma$ applies $\\sigma$ to it.\nThe products $u_\\sigma u_\\tau$ are allowed to differ from $u_{\\sigma\\tau}$ by a nonzero number of $L$, chosen by a rule called a factor set.\nFor $L = \\C$ over $K = \\R$ and the right choice of factor set, the result is Hamilton's quaternions, with $u_\\sigma = j$.\n",
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