  Let $F$ be a $p$-adic field, that is, a finite extension of $\mathbb Q_p$.
Let $D$ be a finite-dimensional central division algebra over $F$ and let
$SL_1(D)$ be the group of elements of reduced norm $1$ in $D$. Prasad and
Raghunathan proved that $H^2(SL_1(D),\mathbb R/\mathbb Z)$ is a cyclic
$p$-group whose order is bounded from below by the number of $p$-power roots of
unity in $F$, unless $D$ is a quaternion algebra over $\mathbb Q_2$. In this
paper we give an explicit upper bound for the order of $H^2(SL_1(D),\mathbb
R/\mathbb Z)$ for $p\geq 5$ and determine $H^2(SL_1(D),\mathbb R/\mathbb Z)$
precisely when $F$ is cyclotomic, $p\geq 19$ and the degree of $D$ is not a
power of $p$.
