the Brauer groups of $\kappa$ and $\kappa(t)$ are uniformly $\ell$-bounded. This result for function fields of $p$-adic curves ($p \neq \ell$) is due to Saltman ([@saltman:jrms]). To prove our theorems we use the patching techniques of Harbater-Hartmann-Krashen and results of ([@ps:period_index]).

We thank D. Harbater for his very useful comments on the text.

 Galois cohomology, Symbol length, $u$-invariant and Uniform bound
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In this section we recall the recent results of Krashen and Saltman connecting the symbol length and effective index in Galois cohomology with the $u$-invariant of a field.

Let $K$ be a field and $\ell$ a prime not equal to the characteristic of $K$. Let $\mu_\ell$ denote the Galois module of $\ell^{\rm th}$ roots of unity and $H^n(K, \mu_\ell^{\otimes m})$ denote the $n^{\rm
  th}$ Galois cohomology group with values in $\mu_\ell^{\otimes m}$. We have $H^1(K, \mu_\ell)  \simeq K^*/K^{*\ell}$. For $a \in K^*$, let $(a) \in H^1(K, \mu_\ell)$ denote the image of $aK^{*\ell}$. Let $a_1, \cdots , a_n \in K^*$. The cup product $(a_1) \cdot (a_2) \cdots (a_n) \in H^n(K, \mu_\ell^{\otimes n})$ is called a [*symbol*]{} in $H^n(K, \mu_\ell^{\otimes n})$. A theorem of Voevodsky ([@voevodsky]) asserts that every element in $H^n(K,
\mu_\ell^{\otimes n})$ is a sum of symbols.

Let 