  Given an element of the Bloch group of a number field~$F$ and a natural
number~$n$, we construct an explicit unit in the field $F_n=F(e^{2 \pi i/n})$,
well-defined up to $\nn$-th powers of nonzero elements of~$F_n$. The
construction uses the cyclic quantum dilogarithm, and under the identification
of the Bloch group of~$F$ with the $K$-group $K_3(F)$ gives \changed{(up to an
unidentified invertible scalar)} a \changed{formula} for a certain abstract
Chern class from~$K_3(F)$. The units we define are conjectured to coincide with
numbers appearing in the quantum modularity conjecture for the Kashaev
invariant of knots (which was the original motivation for our investigation),
and also appear in the radial asymptotics of Nahm sums near roots of unity.
This latter connection is used to prove Nahm's conjecture relating the
modularity of certain $q$-hypergeometric series to the vanishing of the
associated elements in the Bloch group of~$\overline{\mathbb Q}$.
