  In this paper, we continue the study of unit reducible fields as introduced
in \cite{LPL23} for the special case of cyclotomic fields. Specifically, we
deduce that the cyclotomic fields of conductors $2,3,5,7,8,9,12,15$ are all
unit reducible, and show that any cyclotomic field of conductor $N$ is not unit
reducible if $2^4, 3^3, 5^2, 7^2, 11^2$ or any prime $p \geq 13$ divide $N$,
meaning the unit reducible cyclotomic fields are finite in number. Finally, if
$a$ is a totally positive element of a cyclotomic field, we show that for all
equivalent $a^\prime$, the discrepancy between
$\trace_{K/\mathbb{Q}}(a^\prime)$ and the shortest nonzero element of the
quadratic form $\trace_{K/\mathbb{Q}}(axx^*)$ where $x$ is taken from the ring
of integers tends to infinity as the conductor $N$ goes to infinity.
