Period and index

Summary

The period and the index are two numbers attached to a central simple algebra $A$ over a field $K$. The index is the degree of the division algebra $D$ with $A \cong M_m(D)$. The period is the order of the class of $A$ in the Brauer group. The period divides the index, and the two numbers have the same prime divisors. How far apart they can be is the period-index problem, a central question about Brauer groups of fields.