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  "body": "\n## Definition\n\nLet $K$ be a field and let $A$ be a [[central simple algebra|5HRKJBAY2V]] over $K$ of [[degree|NNXSP5KKAC]] $n$.\nLet $L$ be a [[field extension|0EQWVHF3W5]] of $K$.\nThe field $L$ is a *splitting field* of $A$, or *splits* $A$, if there is an isomorphism of $L$-algebras\n$$A \\otimes_K L \\cong M_n(L).$$\nHere $\\otimes_K$ is the [[tensor product of algebras|1735VGRT2P]] and $M_n(L)$ is the [[matrix ring|0FHVPRAE60]] of $n \\times n$ matrices over $L$.\n\n## Remarks\n\nExample: $\\C$ splits $\\mathbb{H}$ over $\\R$.\nIndeed, $\\mathbb{H} \\otimes_\\R \\C$ is central simple of dimension $4$ over $\\C$, and the only such algebra is $M_2(\\C)$, since $\\C$ is algebraically closed.\n\nNon-example: $\\R$ itself does not split $\\mathbb{H}$, since $\\mathbb{H} \\otimes_\\R \\R \\cong \\mathbb{H}$ is a division algebra and $M_2(\\R)$ has zero divisors.\n\n$A$ is split over $K$ itself exactly when $A \\cong M_n(K)$.\nAn algebraic closure of $K$ splits every central simple algebra over $K$, and so does a suitable finite Galois extension of $K$.\nEvery maximal subfield of a central division algebra $D$ is a splitting field of $D$.\n",
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