$E$ be the separable closure of $K$ in $L$. Let $p $ be the characteristic of $(K)$. Suppose $p > 0$. Then $L/E$ is of degree $p^r$ for some $r \geq 0$. Since $\ell \neq
  p$, the restriction map $H^n(E, \mu_\ell^{\otimes m}) \to H^n(L,
  \mu_\ell^{\otimes m})$ is injective ([@serre:gc Cor. on p.12]). Hence $\alpha \otimes_K E = 0$.

We know recall a theorem of Krashen ([@krashen:symbol_length 4.2], cf. [@saltman:symbol_length]).

\[theorem:krashen1\] Let $K$ be a field and $\ell$ a prime not equal to the characteristic of $K$. Let $n \geq 1$. Suppose that there exists an integer $N$ such that for every finite extension $L$ of $K$ and for every element $\beta \in H^d(L, \mu_\ell^{\otimes d})$, $ 1 
  \leq d \leq n-1$, eind$(\beta) \leq N$. Then for any $\alpha \in H^n(K,
  \mu_\ell^{\otimes n})$, $\lambda(\alpha)$ is bounded in term of eind$(\alpha)$, $N$ and $n$.

The following is a consequence of the above theorem.

\[cor:krashen1\] Let $K$ be a field and $\ell$ a prime not equal to the characteristic of $K$. Let $n\geq 1$. Suppose that there exists an integer $N$ such that for all finite extensions $L$ of $K$ and for all $\alpha \in H^d(L, \mu_\ell^{\otimes d})$, $1\leq d \leq
  n$, eind$(\alpha) 