Splitting p-primary cohomology classes of tori in characteristic p

We prove that $p$-primary cohomology classes of a torus $T$ over a global function field of characteristic $p$ may be split by suitable separable $p$-primary extensions. More precisely, we show that such cohomology classes will split in any large'' $p$-primary extension (and in fact, prove the same for $\ell$-primary classes over large’’ $\mathrm{\ell }$-primary extensions for every prime $\mathrm{\ell }$, including $\mathrm{\ell }\mathrm{\ne }\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(K)$), and we prove that $p^n$-torsion classes may be split by a (solvable) separable $p$-primary extension of degree $\le (p^n{)}^{1+cm\mathrm{l}\mathrm{o}\mathrm{g}(m{)}^3}$ for an explicitly computable universal constant $c>0$, where $m$ is the degree of a finite Galois extension splitting the torus $T$. Along the way, we also prove Grunwald-Wang type results of independent interest which allow one to approximate a given finite list of abelian $p$-primary local extensions of places of a global function field by a suitable global extension.