  Full-rate space-time block codes with nonvanishing determinants have been
extensively designed with cyclic division algebras. For these designs, smaller
pairwise error probabilities of maximum likelihood detections require larger
normalized diversity products, which can be obtained by choosing integer
non-norm elements with smaller absolute values. All known methods have
constructed $1+\bi$ and $2+\bi$ to be integer non-norm elements with the
smallest absolute values over QAM for the number of transmit antennas $n$:
$\{n:5\leq n\leq 40,8\nmid n\}$ and $\{n:5\leq n\leq 40,8\mid n\}$,
respectively. Via explicit constructions, this paper proves that $1+\bi$ is an
integer non-norm element with the smallest absolute value over QAM for every
$n\geq 5$.
