  We perform the dimensional reduction of the linear $\sigma$ model at one-loop
level. The effective potential of the reduced theory obtained from the
integration over the nonzero Matsubara frequencies is exhibited. Thermal mass
and coupling constant renormalization constants are given, as well as the
thermal renormalization group equation which controls the dependence of the
counterterms on the temperature. We also recover, for the reduced theory, the
vacuum unstability of the model for large N.
