  In this paper we study the polarization tensor of photons in a QED plasma in
the presence of a magnetic field. We do it both at vanishing and at finite
temperature. We use two different methods to compute the polarization tensor
components $\Pi^{\mu\nu}$ within the one-loop approximation. The first starts
from the effective Lagrangian of the system and relates $\Pi^{\mu\nu}$ to the
thermodynamic quantities. The second makes use of the electron propagator in an
external magnetic field. In this second approach we use an imaginary time
formalism. These methods give consistent results in the first non-vanishing
order in the photon 4-momentum. Beyond this limit the first method is not
applicable and the introduction of the second can not be avoided. We give some
physical interpretations of our results.
