  A PNJL model is built, in which the Polyakov-loop potential is explicitly
Z$_{N_c}$-symmetric in order to mimic a Yang-Mills theory with gauge group
SU($N_c$). The physically expected large-$N_c$ and large-$T$ behaviours of the
thermodynamic observables computed from the Polyakov-loop potential are used to
constrain its free parameters. The effective potential is eventually
U(1)-symmetric when $N_c$ is infinite. Light quark flavours are added by using
a Nambu-Jona-Lasinio (NJL) model coupled to the Polyakov loop (the PNJL model),
and the different phases of the resulting PNJL model are discussed in 't
Hooft's large-$N_c$ limit. Three phases are found, in agreement with previous
large-$N_c$ studies. When the temperature $T$ is larger than some deconfinement
temperature $T_d$, the system is in a deconfined, chirally symmetric, phase for
any quark chemical potential $\mu$. When $T<T_d$ however, the system is in a
confined phase in which chiral symmetry is either broken or not. The critical
line $T_\chi(\mu)$, signalling the restoration of chiral symmetry, has the same
qualitative features than what can be obtained within a standard $N_c=3$ PNJL
model.
