  We present an analytic method to compute the one-loop magnetic correction to
the gluon polarization tensor starting from the Landau-level representation of
the quark propagator in the presence of an external magnetic field. We show
that the general expression contains the vacuum contribution that can be
isolated from the zero-field limit for finite gluon momentum. The general
tensor structure for the gluon polarization also contains two spurious terms
that do not satisfy the transversality properties. However, we also show that
the coefficients of this structures vanish and thus do not contribute to the
polarization tensor, as expected. In order to check the validity of the
expressions we study the strong and weak field limits and show that well
established results are reproduced. The findings can be used to study the
conditions for gluons to equilibrate with the magnetic field produced during
the early stages of a relativistic heavy-ion collision.
