  Given an arbitrary field $\mathbb{F}$ of characteristic 0, we study Lie
bialgebra structures on $sl(n,\mathbb{F})$, based on the description of the
corresponding classical double. For any Lie bialgebra structure $\delta$, the
classical double $D(sl(n,\mathbb{F}),\delta)$ is isomorphic to
$sl(n,\mathbb{F})\otimes_{\mathbb{F}} A$, where $A$ is either
$\mathbb{F}[\varepsilon]$, with $\varepsilon^{2}=0$, or $\mathbb{F}\oplus
\mathbb{F}$ or a quadratic field extension of $\mathbb{F}$. In the first case,
the classification leads to quasi-Frobenius Lie subalgebras of
$sl(n,\mathbb{F})$. In the second and third cases, a Belavin--Drinfeld
cohomology can be introduced which enables one to classify Lie bialgebras on
$sl(n,\mathbb{F})$, up to gauge equivalence. The Belavin--Drinfeld untwisted
and twisted cohomology sets associated to an $r$-matrix are computed. For the
Cremmer--Gervais $r$-matrix in $sl(3)$, we also construct a natural map of sets
between the total Belavin--Drinfeld twisted cohomology set and the Brauer group
of the field $\mathbb{F}$.
