one prime ideal of $B$, $\alpha \otimes F(\sqrt[\ell]{\pi}, \sqrt[\ell]{\delta})$ is unramified on $B$ (cf. \[auslander-goldman\]). The other case follows similarly.

\[theorem:good\_char\_killing\_ramification\] Let $R$ be a complete discrete valuation ring with field of fractions $K$ and residue field $\kappa$. Let $F$ be the function field of a curve over $K$. Let $\ell$ be a prime not equal to char$(\kappa)$ and $\alpha_1, \cdots ,
  \alpha_r \in H^n(K, \mu_\ell^{\otimes m})$. Then there exist $f, g,  h \in F^*$ such that each $\alpha_i \otimes F(\sqrt[\ell]{f}, \sqrt[\ell]{g},
  \sqrt[\ell]{h})$ is unramified at all codimension one points of any regular proper model of $F(\sqrt[\ell]{f}, \sqrt[\ell]{g},
  \sqrt[\ell]{h})$.

Let ${{\mathscr{X}}}$ be a regular proper model of $F$ over $R$ such that the union of the support of ramification locus of $\alpha_i$, for $1
  \leq i \leq r$, is contained in the union of regular curves $C$ and $E$ with $C \cup E$ having only normal crossings. Let $f \in F^*$ be such that $${\rm div}_{{\mathscr{X}}}(f) = C + E + F$$ for some divisor $F$ on ${{\mathscr{X}}}$ which does not contain any irreducible component of $C \cup E$ and does not pass through any point of $C
\cap E$. Let $g \in F^*$ be such that $${\rm div}_{{\mathscr{X}}}(g) = 