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  "body": "\n## Summary\n\nThe period and the index are two numbers attached to a central simple algebra $A$ over a field $K$.\nThe index is the [[degree|NNXSP5KKAC]] of the division algebra $D$ with $A \\cong M_m(D)$.\nThe period is the order of the class of $A$ in the [[Brauer group|BEW48TA9WZ]].\nThe period divides the index, and the two numbers have the same prime divisors.\nHow far apart they can be is the period-index problem, a central question about Brauer groups of fields.\n\n## Explanations\n\n### L1 human\n\nEvery central simple algebra is a matrix algebra over a division algebra, and the index measures the size of that division algebra.\nThe period counts how many copies of the algebra must be tensored together before the result becomes a plain matrix algebra.\nFor Hamilton's quaternions over $\\R$, both numbers are $2$: the quaternions have degree $2$, and $\\mathbb{H} \\otimes_\\R \\mathbb{H}$ is a matrix algebra.\nOver number fields the two numbers are always equal, but over other fields the index can be larger.\n",
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