$P \in {{{\mathscr{P}}}}$. Since $F_P \subset L_y$, by the choices of $h_1, \cdots , h_d, g_1, g_2$, $\alpha_i \otimes L_y = 0$ for all $i$.

Since $[L : F] \leq p^{4d + 4}$, the theorem follows.

\[cor:bad\_char\_2\] Let $K$ be a complete discretely valued field with residue field $\kappa$. Suppose that char$(K) = 0$ and char$(\kappa) = p > 0$ and $[\kappa : \kappa^p] = p^d$. Let $\zeta$ be a primitive $p^{th}$ root of unity. Then $[K(\zeta) : K]p^{4d+4}$ is a $(2, p)$-uniform bound for $K(t)$.

\[cor:bad\_char\_n\] Let $K' = K(\zeta)$. Then $K'$ is a complete discretely valued field with residue field $\kappa$. Let $F$ be a finite extension of $K(t)$. Let $\alpha_1, \cdots , \alpha_m \in H^2(F, \mu_p^{\otimes 2})$. Since $F'= F(\zeta)$ is also a function field over $K'$, by (\[theorem:bad\_char\]), there exists an extension $L$ of $F'$ of degree at most $p^{4d+4}$ such that $\alpha_i \otimes L  = 0$ for all $i$. Since $[L : F] = [L :
  F'][F' : F]$, the corollary follows.

The above corollary and (\[lemma:uniform\_bound\_d\_d+1\]) give the following

Let $K$ be a complete discretely valued field with residue field $\kappa$. Suppose that char$(K) = 0$ and char$(\kappa) = p > 0$ and $[\kappa 