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'' $\ell$-primary extensions for every prime $\ell$, including $\ell \neq \mathrm{char}(K)$), and we prove that $p^n$-torsion classes may be split by a (solvable) separable $p$-primary extension of degree $\leq (p^n)^{1+cm\mathrm{log}(m)^3}$ 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.