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A PNJL model is built, in which the Polyakov-loop potential is explicitlyZ$_{N_c}$-symmetric in order to mimic a Yang-Mills theory with gauge groupSU($N_c$). The physically expected large-$N_c$ and large-$T$ behaviours of thethermodynamic observables computed from the Polyakov-loop potential are used toconstrain its free parameters. The effective potential is eventuallyU(1)-symmetric when $N_c$ is infinite. Light quark flavours are added by usinga 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 'tHooft's large-$N_c$ limit. Three phases are found, in agreement with previouslarge-$N_c$ studies. When the temperature $T$ is larger than some deconfinementtemperature $T_d$, the system is in a deconfined, chirally symmetric, phase forany quark chemical potential $\mu$. When $T<T_d$ however, the system is in aconfined phase in which chiral symmetry is either broken or not. The criticalline $T_\chi(\mu)$, signalling the restoration of chiral symmetry, has the samequalitative features than what can be obtained within a standard $N_c=3$ PNJLmodel.

A large-$N_c$ PNJL model with explicit Z$_{N_c}$ symmetry