  We calculate the structure of a relativistic shock wave in which the internal
energy of the shocked fluid is radiated away on a time scale much shorter than
the characteristic shock propagation time. The shock is assumed to move through
a uniformly magnetized, neutral plasma consisting of protons, electrons, and
positrons, and allowance is made for the possible production of
electron/positron pairs in the shock itself. The radiation mechanism is taken
to be synchrotron and inverse-Compton emission (the latter involving both
synchrotron-produced and externally supplied seed photons) by the electrons and
positrons. We simplify the discussion by considering a shock in which the
magnetic field is transverse to the direction of propagation and focus
attention on the properties of the radiative zone that forms behind the shock
transition. In particular, we investigate the possibility that the compression
induced by the cooling of the gas amplifies the magnetic pressure until it
reaches (and ultimately exceeds) equipartition with the thermal pressure
(which, in turn, limits the overall compression). We show that, if a
significant fraction of the postshock thermal energy is deposited in the
electron/positron component, then a considerable portion of the emitted
radiation will come from regions of strong magnetic field even if the field
immediately behind the shock transition is well below equipartition. This field
amplification mechanism may be relevant to the production of synchrotron flares
in blazars, miniquasars, and gamma-ray burst sources. We consider the latter
application in some detail and show that this process may play a role in the
prompt $\gamma$-ray and possibly also the optical ``flash'' and radio ``flare''
emission, but probably not in the afterglow.
