fractions of $\hat{R}_U$.

Let $A$ be a regular integral domain with field of fractions $F$. For a maximal ideal $m$ of $A$, let $\hat{A}_m$ denote the completion of the local ring $A_m$ and $F_m$ the field of fractions of $\hat{A}_m$.

Uniform bound - bad characteristic case 
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Let $K$ be a complete discretely valued field with residue field $\kappa$. Let $p = char(\kappa)$. In this section we show that there is a $(2, p)$-uniform bound for $K(t)$ which depends only on $[\kappa : \kappa^p]$.

First we recall the following two results from ([@ps:period_index]).

\[theorem:ps1\] [@ps:period_index 2.4] Let $R$ be a complete discrete valuation ring with field of fractions $K$ and residue field $\kappa$. Suppose that char$(K) = 0$, char$(\kappa)
  = p >0$ and $[\kappa : \kappa^p] = p^d$. Let $\pi \in R$ be a parameter and $u_1,\cdots ,u_d \in R^*$ units such that $\kappa =
  \kappa^p(\overline{u}_1, \cdots , \overline{u}_d)$, where for any $u \in R$, $\overline{u}$ denotes the image of $u$ in $\kappa$. Suppose that $K$ contains a primitive $p^{\rm th}$ root of unity. Then any $\alpha \in H^2(K, \mu_p) $ splits over $K(\sqrt[p]{\pi}, \sqrt[p^2]{u_1}, \cdots ,  \sqrt[p^2]{u_{d-1}}, \sqrt[p]{u_d})$. In particular if $d > 0$, $p^{2d}$ is a $(2, p)$-uniform 