: \kappa^p] = p^d$. Let $\zeta$ be a primitive $p^{th}$ root of unity. Then $[K(\zeta) : K]p^{4d+4}$ is an $(n, p)$-uniform bound for $K(t)$ for all $n \geq 2$.

Uniform bound - good characteristic case 
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Let $F$ be the function field of a $p$-adic curve. In ([@saltman:jrms]), Saltman proved that if $\ell$ is a prime not equal to $p$ and $\alpha_1, \cdots , \alpha_m \in H^2(F, \mu_\ell)$, then there exists an extension $L$ of $F$ such that $[L : F] \leq \ell^2$ and $\alpha_{i_L} = 0 $ for all $i$, i.e., $F$ is $(2,
\ell)$-uniformly bounded.

Let $K$ be a complete discretely valued field with residue field $\kappa$. Let $\ell$ be a prime not equal to char$(\kappa)$. In this section we show that $K(t)$ is $(2,\ell)$-uniformly bounded under some conditions on $\kappa$.

\[theorem:good\_char\_dvr\] Let $K$ be a complete discretely valued field with residue field $\kappa$. Let $\ell$ be a prime not equal to char$(\kappa)$ and $n \geq 1$. If $N$ is an $(n, \ell)$-uniform bound for $\kappa$, then $\ell N$ is an $(n, \ell)$-uniform bound for $K$.

Let $L$ be a finite extension of $K$. Then $L$ is a complete discretely valued field with residue $\kappa'$ a finite extension 