  We construct unramified central simple algebras representing 2-torsion
classes in the Brauer group of a hyperelliptic curve, and show that every
2-torsion class can be constructed this way when the curve has a rational
Weierstrass point or when the base field is C_1. In general, we show that a
large (but in general proper) subgroup of the 2-torsion classes are given by
the construction. Examples demonstrating applications to the arithmetic of
hyperelliptic curves defined over number fields are given.
