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  "body": "\n## Summary\n\nLet $A$ be a [[central simple algebra|5HRKJBAY2V]] over a field $K$.\nA field extension $L$ of $K$ splits $A$ if $A$ becomes a full matrix algebra after extending scalars to $L$.\nEvery central simple algebra has a splitting field of finite degree over $K$, and even one that is a finite Galois extension of $K$.\nSplitting fields link central simple algebras with Galois cohomology: the algebras split by $L$ give the relative Brauer group $\\mathrm{Br}(L/K)$.\nThe word is unrelated to the [[splitting field|0KXQKX9P63]] of a polynomial, apart from the idea of making something decompose.\n\n## Explanations\n\n### L1 human\n\nA central simple algebra may be a division algebra over a small field, yet look like an ordinary matrix algebra over a larger one.\nA splitting field is a larger field that makes this happen.\nFor example, Hamilton's quaternions are a division algebra over $\\R$, but over $\\C$ they become the algebra of $2 \\times 2$ complex matrices, so $\\C$ splits them.\n",
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