  We introduce the formalism of quantum cosmology in a
Friedmann-Lema\^itre-Robertson-Walker (FLRW) universe of arbitrary dimension
filled with a perfect fluid with $p=\alpha\rho$ equation of state. First we
show that the Schutz formalism, developed in four dimensions, can be extended
to a n-dimensional universe. We compute the quantum representant of the scale
factor $a(t)$, in the Many-Worlds, as well as, in the de Broglie-Bohm
interpretation of quantum mechanics. We show that the singularities, which are
still present in the n-dimensional generalization of FLRW universe, are
excluded with the introduction of quantum theory. We quantize, via the de
Broglie-Bohm interpretation of quantum mechanics, the components of the Riemann
curvature tensor in a tetrad basis in a n-dimensional FLRW universe filled with
radiation ($p=\frac{1}{n-1}\rho$). We show that the quantized version of the
Ricci scalar are perfectly regular for all time $t$. We also study the behavior
of the energy density and pressure and show that the ratio $<p>_L/<\rho>_L$
tends to the classical value $1/(n-1)$ only for $n=4$, showing that $n=4$ is
somewhat privileged among the other dimensions. Besides that, as $n\to\infty$,
$<p>_L/<\rho>_L\to 1$.
