\leq N$. Then there exists an integer $N'$ which depends only $N$ and $n$ such that $\lambda(\alpha) \leq N'$ for all finite extensions $L$ of $K$ and $\alpha
  \in H^n(K, \mu_\ell^{\otimes n})$.

Let $K$ be a field of characteristic not equal to 2. The [*$u$-invariant*]{} of $K$ is defined to be the supremum of dimensions of anisotropic quadratic forms over $K$. The following theorem is a consequence of a theorem of Orlov, Vishik, and Voevodsky ([@ovv]) on the Milnor conjecture (cf. [@kahn], [@ps:symbol_length]).

\[theorem:symbol\_length\_u\_invariant\] Let $K$ be a field of characteristic not equal to 2. Suppose that there exist integer $M\geq 1 $ and $N$ such that $H^M(K,
\mu_2) =0$ and $\lambda(\alpha) \leq N$ for all $\alpha \in H^d(K,
\mu_2^{\otimes d})$, $ 1 \leq d < M$. Then the $u$-invariant is bounded by a function of $M$ and $N$.

The following follows from (\[theorem:symbol\_length\_u\_invariant\]) and (\[cor:krashen1\]) (cf. [@krashen:symbol_length 5.5]).

\[cor:krashen2\] Let $K$ be a field of characteristic not equal to 2. Suppose that there exist integers $N \geq 1$ and $M \geq
1$ such that for all finite extensions $L$ of $K$, $H^{M}(L,
\mu_2) = 0$ and for all $ n \geq 1$ and $\alpha \in H^n(L, \mu_2)$, eind$(\alpha) \leq 