  We investigate the effect of a strong magnetic field on dynamical chiral
symmetry breaking in quenched and unquenched QCD. To this end we apply the
Ritus formalism to the coupled set of (truncated) Dyson-Schwinger equations for
the quark and gluon propagator under the presence of an external constant
Abelian magnetic field. We work with an approximation that is trustworthy for
large fields eH > \Lambda_{QCD}^2 but is not restricted to the lowest Landau
level. We confirm the linear rise of the quark condensate with large external
field previously found in other studies and observe the transition to the
asymptotic power law at extremely large fields. We furthermore quantify the
validity of the lowest Landau level approximation and find substantial
quantitative differences to the full calculation even at very large fields. We
discuss unquenching effects in the strong field propagators, condensate and the
magnetic polarization of the vacuum. We find a significant weakening of
magnetic catalysis caused by the back reaction of quarks on the Yang-Mills
sector. Our results support explanations of the inverse magnetic catalysis
found in recent lattice studies due to unquenching effects.
