Submission: 2009, Nov 5, revised: 2009, Nov 13
We associate to any central simple algebra $A$ of exponent $2$ over a field of characteristic $\ne 2$ an invariant with values in the degree $2$ unramified cohomology of its Severi-Brauer variety modulo the image of the cohomology of the ground field. The main theorem is that this invariant is nonzero if and only if the index of $A$ is $\ge 8$.
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