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answered Is any quadric birational to a product of Brauer-Severi varieties?
Oct
24
awarded  Supporter
Oct
24
comment Brauer group elements of order $2$
And to expand on this, if $F$ is any field of characteristic not 2, and one considers the purely transcendental extension $F(x, y, z, w)$, then over this field, the product of the two quaternion algebras: $(x,y) \otimes__{F(x,y,z,w)} (z,w)$ is a division algebra, by Albert's criteria above, and hence has index 4. It has period two because each quaternion algebra does.
Sep
16
awarded  Teacher
Sep
16
answered Infinite dimensional central simple algebras