extension of $K(t)$. Let $\alpha_1, \cdots , \alpha_m \in H^2(F,\mu_p)$. Let ${{{\mathscr{X}}}}$ be a regular proper model of $F$ over the ring of integers $R$ of $K$ such that the support of ram$(\alpha_i)$ for all $i$ and the special fiber is contained in $C \cup E$, where $C$ and $E$ are regular curves on ${{\mathscr{X}}}$ having only normal crossings. Let $\eta$ be the generic point of an irreducible component $X_\eta$ of the special fiber of ${{\mathscr{X}}}$. Then $\kappa(\eta)$ is a function field in one variable over $\kappa$. Since $[\kappa : \kappa^p] = p^d$, $[\kappa(\eta) : \kappa(\eta)^p] =
d+1$ ([@bourbaki:algebra; @ll A.V.135, Corollary 3]). Let $\pi_\eta$ be a parameter at $\eta$ and $u_{\eta, 1},  \cdots ,  u_{\eta, d+1} \in F^*$ be lifts of a $p$-basis of $\kappa(\eta)$. Then, by (\[theorem:ps1\]), $\alpha \otimes F_\eta(\sqrt[p^2]{u_{\eta, 1}}, 
\sqrt[p^2]{u_{\eta, d}}, \sqrt[p]{u_{\eta, d+1}}, \sqrt[p]{\pi_\eta}) = 0$ for all $i$. Let $f \in F^*$ be chosen such that $\nu_\eta(f) = 1$ for all $\eta$. By the Chinese remainder theorem, choose $u_1,  \cdots , u_d \in F^*$ units at each $\eta$ such that $\overline{u}_j = 
\overline{u}_{\eta, j} \in
\kappa(\eta)$. Then $\alpha_i \otimes F_\eta(\sqrt[p]{f}, 
 \sqrt[p^2]{u_1}, \cdots , \sqrt[p^2]{u_{d}}, \sqrt[p]{u_{d+1}}) = 0$ for all $i$. By ([@hhk:lgp_torsors 5.8], [@kmrt 1.17]), 