of $\kappa$. Let $R$ be the valuation ring of $R$ and $\pi \in R$ be a parameter. Let $\alpha_1, \cdots, \alpha_m \in H^n(L,
  \mu_\ell^{\otimes n})$. Let $S$ be the integral closure of $R$ in $L(\sqrt[\ell]{\pi})$. Then $S$ is also a complete discrete valuation ring with residue field $\kappa'$. Since $S/R$ is ramified, $\alpha_i \otimes
  L(\sqrt[\ell]{\pi})$ is unramified at $S$ for each $i$. Hence there exists $\beta_i \in H^n_{{\mathrm{\acute{e}t}}}(S, \mu_\ell^{\otimes n})$ such that $\beta_i
  \otimes_SK(\sqrt[\ell]{\pi}) = \alpha_i$. Since $N$ is an $(n,
  \ell)$-uniform bound for $\kappa$, there exists an extension $L_0$ of $\kappa$ of degree at most $N$ such that $\beta_i \otimes_\kappa L_0 =
  0$ for all $i$. Let $L$ be the extension of $K$ of degree equal to $[L_0 : \kappa]$ with residue field $L_0$. Let $T$ be the integral closure of $R$ in $L(\sqrt[\ell]{\pi})$. Then $T$ is a complete discrete valued ring with residue field $L_0$ and $S \subset T$. Since $\beta_i \otimes_\kappa L_0 = 0$, $\beta_i
  \otimes_S T = 0$ for all $i$. In particular $\alpha_i \otimes_K
  L(\sqrt[\ell]{\pi}) = 0$ for all $i$. Since the degree of $L$ over $K$ is equal to the degree of $L_0$ over $\kappa$ and $[L_0 :
  \kappa] \leq N$, we have $[L(\sqrt[\ell]{\pi}) 