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  "body": "\n## Definition\n\nLet $K$ be a field and let $A$ be a [[central simple algebra|5HRKJBAY2V]] over $K$.\nBy Wedderburn's theorem $A \\cong M_m(D)$ for an integer $m \\ge 1$ and a [[division ring|0EA5BTA0CX]] $D$ that is central simple over $K$, unique up to isomorphism.\n\n1. The *index* of $A$, written $\\operatorname{ind} A$, is the [[degree|NNXSP5KKAC]] of $D$.\n2. The *period* (or *exponent*) of $A$, written $\\operatorname{per} A$, is the [[order|0WE2NHZJ7B]] of the class $[A]$ in the [[Brauer group|BEW48TA9WZ]] $\\mathrm{Br}(K)$.\n\n## Remarks\n\nBasic facts: $\\operatorname{per} A$ divides $\\operatorname{ind} A$, $\\operatorname{ind} A$ divides $\\deg A$, and $\\operatorname{per} A$ and $\\operatorname{ind} A$ have the same prime divisors.\nIn particular the period is finite, so $\\mathrm{Br}(K)$ is a torsion group.\n\nExample: for $\\mathbb{H}$ over $\\R$, $D = \\mathbb{H}$ has degree $2$, so the index is $2$.\nThe class $[\\mathbb{H}]$ is not trivial, and $\\mathbb{H} \\otimes_\\R \\mathbb{H} \\cong M_4(\\R)$, so the period is $2$.\n\nExample: $A = M_n(K)$ has $D = K$, so its index and its period are both $1$.\n\nOver local fields and over global fields, the period equals the index for every central simple algebra.\nOver general fields the index can be strictly larger than the period.\n",
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