To study the nuclear symmetry energy, we extend the Dirac-Brueckner ap
proach with a Bonn one-boson-exchange nucleon-nucleon interaction to t
he general case of asymmetric nuclear matter. We extract the symmetry
energy coefficient at the saturation to be about 31 MeV, which is in g
ood agreement with the empirical value of 30+/-4 MeV. The symmetry ene
rgy is found to increase almost linearly with the density, which diffe
rs considerably from the results of nonrelativistic approaches. This f
inding also supports the linear parametrization of Prakash, Ainsworth,
and Lattimer. We find, furthermore, that the higher-order dependence
of the nuclear equation of state on the asymmetry parameter is unimpor
tant.