  The accretion flow around X-ray pulsars with a strong magnetic field is
funnelled by the field to relatively small regions close to the magnetic poles
of the neutron star (NS), the hotspots. During strong outbursts regularly
observed from some X-ray pulsars, the X-ray luminosity can be so high, that the
emerging radiation is able to stop the accreting matter above the surface via
radiation-dominated shock, and the accretion column begins to rise. This border
luminosity is usually called the "critical luminosity". Here we calculate the
critical luminosity as a function of the NS magnetic field strength $B$ using
exact Compton scattering cross section in strong magnetic field. Influence of
the resonant scattering and photon polarization is taken into account for the
first time. We show that the critical luminosity is not a monotonic function of
the B-field. It reaches a minimum of a few 10^{36} erg s^{-1} when the
cyclotron energy is about 10 keV and a considerable amount of photons from a
hotspot have energy close to the cyclotron resonance. For small B, this
luminosity is about 10^{37} erg s^{-1}, nearly independent of the parameters.
It grows for the B-field in excess of 10^{12} G because of the drop in the
effective cross-section of interaction below the cyclotron energy. We
investigate how different types of the accretion flow and geometries of the
accretion channel affect the results and demonstrate that the general behaviour
of the critical luminosity on B-field is very robust. The obtained results are
shown to be in a good agreement with the available observational data and
provide a necessary ground for the interpretation of upcoming high quality data
from the currently operating and planned X-ray telescopes.
