bound for $K$ and if $d = 0$, $p$ is a $(2, p)$-uniform bound for $K$.

\[prop:ps2\] ([@ps:period_index 3.5]) Let $A$ be a complete regular local ring of dimension 2 with field of fractions $F$ and residue field $\kappa$. Suppose that char$(F) =
  0$, char$(\kappa) = p > 0$ and $[\kappa : \kappa^p] = p^d$. Let $\pi, \delta \in A$ and $u_1, \cdots , u_d \in A^*$ such that the maximal ideal $m$ of $A$ is generated by $\pi$ and $\delta$, and $\kappa = \kappa^p(\overline{u}_1, \cdots ,
\overline{u}_d)$, where for any $u \in A$, $\overline{u}$ denotes the image of $u$ in $\kappa$. Suppose that $F$ contains a primitive $p^{\rm th}$ root of unity. Then any $\alpha \in H^2(F, \mu_p)$ which is unramified on $A$ except possibly at $(\pi)$ and $(\delta)$ splits over $F(\sqrt[p]{\pi},  \sqrt[p]{\delta}, \sqrt[p^2]{u_1},
  \cdots ,   \sqrt[p^2]{u_d})$.

\[lemma:choice\_of\_f\_g\] Let ${{\mathscr{X}}}$ be a regular integral two dimensional scheme and $F$ its function field. Suppose $C$ and $E$ are regular curves on ${{\mathscr{X}}}$ with normal crossings. Let ${{\mathscr{P}}}$ be a finite set of closed points of ${{\mathscr{X}}}$ with ${{\mathscr{P}}}\subset C \cup E$. Then there exist $f, g \in F^*$ such that the maximal ideal at $P$ is generated by $f$ and 