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Let $k$ be a field and $l$ be a prime, coprime to the characteristic of $k$. Suppose
that $k$ contains the $l$-th roots of unity. Every element in the Galois cohomology
$H^n(k, \mu_l)$ is a sum of ``symbols'', thanks to a theorem of Voevodsky. It is interesting
to get bounds $N(n)$ such that every such element is a sum of at most $N(n)$ symbols,
for fields of arithmetic interest. Using recent results of Saltman, we show that if $k$
is a $p$-adic field and $p$ not equal to $l$ and $K$ is the function field of a curve over $k$,
then, $N(2) = 2$ and $N(3) = 1$.
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