: K] \leq \ell N$.

\[lemma:good\_char\_dim2\] Let $A$ be a regular local ring with residue field $\kappa$ and maximal ideal $m = (\pi,\delta)$. Let $F$ be the field of fractions of $A$, $\ell$ a prime not equal to char$(\kappa)$. Let $B$ (resp. $B'$) be the integral closure of $A$ in $F(\sqrt[\ell]{\pi}, \sqrt[\ell]{\delta})$ (resp. $F(\sqrt[\ell]{\pi})$). Let $\alpha \in H^2(F, \mu_\ell^{\otimes 2})$. If $\alpha$ is unramified on $A$ except possibly at $(\pi)$ and $(\delta)$ (resp. except possibly at $(\pi)$), then $\alpha \otimes_F F(\sqrt[\ell]{\pi},
\sqrt[\ell]{\delta})$ is unramified on $B$ (resp. $\alpha \otimes_F
F(\sqrt[\ell]{\pi})$ is unramified on $B'$).

By ([@ps:period_index 3.3]), $B$ is a regular local ring of dimension 2 with residue field $\kappa$. Let $P$ be a height one prime ideal of $B$ and $Q = P \cap A$. Then $Q$ is a height one prime ideal of $A$. If $Q \neq (\pi)$ and $(\delta)$, then $\alpha$ is unramified at $Q$ and hence $\alpha \otimes F(\sqrt[\ell]{\pi}, \sqrt[\ell]{\delta})$ is unramified at $P$. Suppose $Q = (\pi)$ or $(\delta)$. Then $Q$ is ramified in $B$ and hence $\alpha \otimes F(\sqrt[\ell]{\pi}, \sqrt[\ell]{\delta})$ is unramified at $P$. Since $B$ is a regular local ring of dimension 2 and $\alpha \otimes F(\sqrt[\ell]{\pi}, \sqrt[\ell]{\delta})$ is unramified at every height 