  In this paper, we present two main results. Let $K$ be a number field that is
Galois over $\mathbb{Q}$ with degree $r+2s$, where $r$ is the number of real
embeddings and $s$ is the number of pairs of complex embeddings. The first
result states that the number of facets of the reduction domain (and therefore
the fundamental domain) of $K$ is no greater than
$O\left(\left(\frac{1}{2}(r+s-1)^\delta(r+s)^{1+\frac{1}{2(r+s-1)}}\right)^{r+s-1}\right)
\cdot\left(e^{1+\frac1{2e}}\right)^{r+s}(r+s)!$, where $\delta=1/2$ if $r+s
\leq 11$ or $\delta=1$ otherwise. The second result states that there exists a
linear time algorithm to reduce a totally positive unary form $axx^*$, such
that the new totally positive element $a^\prime$ that is equivalent to $a$ has
trace no greater than a constant multiplied by the integer minimum of the
trace-form $\trace(axx^*)$, where the constant is determined by the shortest
Pisot unit in the number field. This may have applications in ring-based
cryptography. Finally, we show that the Weil height of the shortest Pisot unit
in the number field can be no greater than
$\frac{1}{[K:\mathbb{Q}]}\left(\frac{\gamma}{2}(r+s-1)^{\delta-\frac{1}{2(r+s-1)}}R_K^{\frac{1}{r+s-1}}+(r+s-1)\epsilon\right)$,
where $R_K$ denotes the regulator of $K$, $\gamma=1$ if $K$ is totally real or
$2$ otherwise, and $\epsilon>0$ is some arbitrarily small constant.
