The linear Z(3) spin system in the presence of a magnetic field is studied by a dedecoration renormalisation group (RG) transformation. In order to perform the transformation, non-usual terms must be added to the Hamiltonian. The linear region around a particular T=0 fixed point, where these terms go to zero, is investigated and the 'critical behaviour' of the free energy, correlation function and susceptibility is obtained. The system is solved exactly and the results are compared with those obtained by the RG approach. Magnetisation and susceptibility curves against temperature are also shown.