N$. Then there exists an integer $N'$, which depends only on $N$ and $M$, such that $u(K)  \leq N'$.

Let $K$ be a field and $\ell$ a prime not equal to the characteristic of $K$. We say that $K$ is $(n , \ell)$-[*uniformly bounded* ]{} if there exists an integer $N$ such that for any finite extension $L$ of $K$ and $\alpha_1, \cdots , \alpha_m
\in H^n(L, \mu_\ell^{\otimes   n})$ there is an extension $E$ of $L$ with $[E : L] \leq N$ and $\alpha_{i} \otimes_L E = 0$ for $1 \leq i \leq m$. Such an $N$ is called an $(n, \ell)$-[*uniform bound*]{} for $K$. We note that if $N$ is an $(n, \ell)$-uniform boud for $K$, then $N$ is also an $(N, \ell)$-uniform bound for any finite extension $L$ of $K$.

In view of a theorem of Voevodsky ([@voevodsky]) on the Bloch-Kato conjecture, every element in $H^n(K, \mu_\ell^{\otimes n})$ is a sum of symbols. In particular $N$ is an $(n, \ell)$-uniform bound of $K$ if and only if for given symbols $\alpha_1, \cdots , \alpha_m \in H^n(K, \mu_\ell^{\otimes
  n})$ there is an extension $L$ of $K$ with $[L : K] \leq N$ and $\alpha_{i_L} = 0$ for $1 \leq 