$K(t)$, $u(F) \leq M$.

It was conjectured in ([@ps:period_index]) that $u(F)$ is at most $8[\kappa: \kappa^2]$. The bound we give for the $u(F)$ is not effective and far from the conjectural bound.

Let $L$ be a field of characteristic not equal to 2 with $H^M(L,
\mu_2^{\otimes M}) = 0$ for some $M \geq 1$. Suppose that there exists an integer $N$ such that for all finite extensions $E$ of $L$ and for any $\alpha \in H^n(E, \mu_2^{\otimes n}), n \geq 2$, there exists an extension $E'$ of $E$ of degree at most $N$ such that $\alpha  \otimes_E E' = 0$. Then a theorem of Krashen (\[cor:krashen2\]) asserts that the $u$-invariant of $L$ is finite. Our aim is to prove that if $K$ is a complete discretely valued field with residue field $\kappa$ of characteristic 2 and $[\kappa : \kappa^2]$ finite and $F$ a function field in one variable over $K$, then such an integer $N$ exists for $F$, thereby proving the finiteness of the $u$-invariant of $F$.

We introduce the notion of uniform boundedness for the $\ell$-torsion of the Brauer group $\operatorname{\mathrm{Br}}(L)$ of $L$, where $L$ is any field. We say that the Brauer group of $L$ is [*uniformly $\ell$-bounded*]{} 