  This paper explores the effects of strong magnetic fields on the Compton
scattering of relativistic electrons. Recent studies of upscattering and energy
loss by relativistic electrons that have used the non-relativistic, magnetic
Thomson cross section for resonant scattering or the Klein-Nishina cross
section for non-resonant scattering do not account for the relativistic quantum
effects of strong fields ($ > 4 \times 10^{12}$ G). We have derived a
simplified expression for the exact QED scattering cross section for the
broadly-applicable case where relativistic electrons move along the magnetic
field. To facilitate applications to astrophysical models, we have also
developed compact approximate expressions for both the differential and total
polarization-dependent cross sections, with the latter representing well the
exact total QED cross section even at the high fields believed to be present in
environments near the stellar surfaces of Soft Gamma-Ray Repeaters and
Anomalous X-Ray Pulsars. We find that strong magnetic fields significantly
lower the Compton scattering cross section below and at the resonance, when the
incident photon energy exceeds $m_ec^2$ in the electron rest frame. The cross
section is strongly dependent on the polarization of the final scattered
photon. Below the cyclotron fundamental, mostly photons of perpendicular
polarization are produced in scatterings, a situation that also arises above
this resonance for sub-critical fields. However, an interesting discovery is
that for super-critical fields, a preponderance of photons of parallel
polarization results from scatterings above the cyclotron fundamental. This
characteristic is both a relativistic and magnetic effect not present in the
Thomson or Klein-Nishina limits.
