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  "body": "\n## Definition\n\nLet $L/K$ be a finite [[Galois extension|0QMZAQ6QQ4]] with [[Galois group|0XTY86BZKY]] $G$ and degree $n = [L : K]$.\nA *factor set* (or $2$-cocycle) is a map $f \\colon G \\times G \\to L^\\times$ such that\n$$f(\\sigma, \\tau) \\, f(\\sigma\\tau, \\rho) = \\sigma\\big(f(\\tau, \\rho)\\big) \\, f(\\sigma, \\tau\\rho) \\quad \\text{for all } \\sigma, \\tau, \\rho \\in G.$$\nThe *crossed product algebra* $(L/K, f)$ is the $L$-vector space with basis $\\set{u_\\sigma : \\sigma \\in G}$, with the multiplication determined by\n$$u_\\sigma \\, x = \\sigma(x) \\, u_\\sigma \\quad (x \\in L), \\qquad u_\\sigma \\, u_\\tau = f(\\sigma, \\tau) \\, u_{\\sigma\\tau}.$$\n\n## Remarks\n\nThe cocycle condition is exactly what makes the multiplication associative.\nThe algebra $(L/K, f)$ is a [[central simple algebra|5HRKJBAY2V]] over $K$ of degree $n$, it contains $L$ as a maximal subfield, and $L$ splits it.\nTwo factor sets give isomorphic algebras if and only if they differ by a coboundary, so the classes correspond to $H^2(G, L^\\times)$ in [[Galois cohomology|9MPCZ3EAJM]].\nThis gives $\\mathrm{Br}(L/K) \\cong H^2(G, L^\\times)$.\n\nExample: take $L = \\C$, $K = \\R$, $G = \\set{1, \\sigma}$ with $\\sigma$ complex conjugation, and $f(\\sigma, \\sigma) = -1$, $f = 1$ otherwise.\nWith $j = u_\\sigma$ one gets $j z = \\bar{z} j$ and $j^2 = -1$, which are the relations of $\\mathbb{H} = \\C + \\C j$.\nWith $f(\\sigma, \\sigma) = 1$ instead one gets an algebra isomorphic to $M_2(\\R)$.\n\nNon-example: for $L = \\C$, $K = \\R$ as above, the map with $f(\\sigma, \\sigma) = i$ and $f = 1$ otherwise is not a factor set.\nThe condition for $\\sigma, \\sigma, \\sigma$ would give $i = \\sigma(i) = -i$, and indeed $(u_\\sigma u_\\sigma) u_\\sigma = i u_\\sigma$ while $u_\\sigma (u_\\sigma u_\\sigma) = \\sigma(i) u_\\sigma = -i u_\\sigma$, so the product is not associative.\n\nA central simple algebra is called a crossed product if it is isomorphic to some $(L/K, f)$; Amitsur showed that some central division algebras are not crossed products.\n",
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