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  "body": "\n## Summary\n\nA quaternion algebra over a field $F$ is a [[central simple algebra|5HRKJBAY2V]] over $F$ of dimension $4$.\nWhen the characteristic of $F$ is not $2$, it has a basis $1, i, j, k$ with $i^2 = a$, $j^2 = b$ and $ij = -ji = k$ for some nonzero $a, b \\in F$.\nEach one is either a division algebra or isomorphic to the algebra of $2 \\times 2$ matrices over $F$.\nThe class of a quaternion algebra in the Brauer group has order at most $2$.\nQuaternion algebras are tied to quadratic forms and the Hilbert symbol.\n\n## Explanations\n\n### L1 human\n\nHamilton's [[quaternions|0GE4DVQKR8]] are numbers $x + yi + zj + wk$ with $i^2 = j^2 = -1$ and $ij = -ji = k$.\nA quaternion algebra keeps this pattern but allows other values for $i^2$ and $j^2$ and other fields of coefficients.\nWith $i^2 = j^2 = -1$ over $\\R$ one gets Hamilton's quaternions, a division algebra.\nWith $i^2 = j^2 = 1$ over $\\R$ one gets an algebra isomorphic to the $2 \\times 2$ real matrices, which has zero divisors such as $1 + i$.\n",
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