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  "body": "\n## Counterexample\n\nLet $k = \\Q$ and let $n \\ge 2$.\nLet $X$ and $Y$ be two $n \\times n$ matrices whose entries are independent commuting variables over $k$.\nThe $k$-algebra generated by $X$ and $Y$ is the [[generic matrix algebra|GJGEVMZ828]]; it has no zero divisors.\nIts ring of central quotients $\\mathrm{UD}(k, n)$ is the *generic division algebra*.\nIt is a [[division ring|0EA5BTA0CX]] and a [[central simple algebra|5HRKJBAY2V]] of [[degree|NNXSP5KKAC]] $n$ over its center.\nAmitsur proved that $\\mathrm{UD}(k, n)$ is not a [[crossed product algebra|G21MZQCME9]] when $n$ is divisible by $8$ or by $p^2$ for an odd prime $p$.\nThat is, for such $n$ it contains no maximal subfield that is a Galois extension of its center.\n\n## Remarks\n\nThe idea of the proof is to specialize.\nA crossed product structure on $\\mathrm{UD}(k, n)$ with group $G$ would pass to every division algebra of degree $n$ over suitable fields of iterated Laurent series.\nAmitsur exhibited two such division algebras that are crossed products only for groups of incompatible types, so no single group $G$ works.\nFor small degrees the opposite holds: every central division algebra of degree $2$, $3$, $4$ or $6$ is a crossed product.\nOver local and global fields every central simple algebra is cyclic, so noncrossed products can only appear over more complicated fields.\n\n## Explanations\n\n### L1 human\n\nFor a long time it was open whether every finite-dimensional central division algebra comes from the crossed product recipe.\nAmitsur found division algebras where this fails.\nThey are built from two \"generic\" square matrices of variables, so that they carry no special structure that a crossed product would need.\n",
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