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An extension $E/K$ splits a central simple algebra $A$ of degree $d$ when $A{\otimes }_{K}E\cong M_d(E)$. A finite separable splitting field exists. Splitting is preserved by further scalar extension and is equivalent to the vanishing of the restricted Brauer group class.

The function field of a curve over an algebraically closed field is a C1 field. Consequently its Brauer group is zero. This is the input for cyclic splitting by extension of constants over a finite constant field.