Cyclic splitting by extension of constants
= Cyclic splitting by extension of constants
For the function field $K$ of a curve over a finite full constant field $k$, extend constants to $\overline k$. The resulting curve function field is $C_1$ by the <Tsen theorem>, so every <central simple algebra> splits there. The finite collection of coefficients of a splitting isomorphism descends to some finite constant extension $k^{\prime}/k$. Regularity makes $Kk^{\prime}/K$ cyclic, giving a finite cyclic splitting field.