  The chiral phase transition is investigated within the framework of the
linear sigma model at finite temperature. We concentrate on the meson sector of
the model and calculate the finite temperature effective potential in the
Hartree approximation by using the Cornwall-Jackiw-Tomboulis formalism of
composite operators. The effective potential is calculated for N=4 involving
the usual sigma and three pions and in the large N approximation involving N-1
pion fields. In the N=4 case we have examined the theory both in the chiral
limit and with the presence of a symmetry breaking term which generates the
pion masses. In both cases we have solved the system of the resulting gap
equations for the thermal effective masses of the particles numerically and we
have investigated the evolution of the effective potential. In the N=4 case
there is indication of a first order phase transition and the Goldstone theorem
is not satisfied. The situation is different in the general case using the
large $N$ approximation, the Goldstone theorem is satisfied and the phase
transition is of the second order. For this analysis we have ignored quantum
effects and we used the imaginary time formalism for calculations.
