Period and index

Remarks

Basic 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. In particular the period is finite, so $\mathrm{Br}(K)$ is a torsion group.

Example: for $\mathbb{H}$ over $\R$, $D = \mathbb{H}$ has degree $2$, so the index is $2$. The class $[\mathbb{H}]$ is not trivial, and $\mathbb{H} \otimes_\R \mathbb{H} \cong M_4(\R)$, so the period is $2$.

Example: $A = M_n(K)$ has $D = K$, so its index and its period are both $1$.

Over local fields and over global fields, the period equals the index for every central simple algebra. Over general fields the index can be strictly larger than the period.