$\alpha \in H^n(K,
\mu_\ell^{\otimes n})$. The [*symbol length*]{} of $\alpha$, denoted by $\lambda(\alpha)$, is defined as the smallest $m$ such that $\alpha $ is a sum of $m$ symbols in $H^n(K, \mu_\ell^{\otimes n})$. For any $\alpha \in H^n(K, \mu_\ell^{\otimes m})$, the [ *effective index*]{} of $\alpha$, denoted by $eind(\alpha)$, is defined to be the minimum of the degrees of finite field extensions $E$ of $K$ with $\alpha_E = 0$, where $\alpha_E$ is the image of $\alpha$ in $H^n(E, \mu_\ell^{\otimes m})$. Since $H^2(K, \mu_\ell) \simeq
{{}_\ell\mathrm{Br}}(K)$, for $\alpha \in {{}_\ell\mathrm{Br}}(K)$, eind$(\alpha)$ is equal to the index of a central simple algebra $A$ over $K$ representing $\alpha$. The following lemma asserts that this definition of effective index coincides with the definition in ([@krashen:symbol_length]).

\[lemma:sep\_ext\] Let $K$ be a field and $\ell$ a prime not equal to char$(K)$. Let $\alpha \in H^n(K, \mu_\ell^{\otimes
    m})$. Suppose that exists an extension $L$ of $K$ of degree at most $N$ with $\alpha \otimes_K L = 0$. Then there exists a separable field extension $E$ of $K$ of degree at most $N$ such that $\alpha
  \otimes_K E = 0$.

Let $L$ be an extension of $K$ of degree at most $N$ with $\alpha \otimes_KL  = 0$. Let 