  We study the synchrotron and synchrotron self-Compton (SSC) emission from
internal shocks that are responsible for the prompt gamma-ray emission in GRBs,
and consider the relation between these two components, taking into account the
high energy (HE) cutoff due to pair production and Thomson scattering. We find
that in order for the peak energy of the synchrotron to be E_p\sim 300 keV with
a variability time t_v>1 ms, a Lorentz factor \Gamma<350 is needed, implying no
HE emission above \sim 30 MeV and the synchrotron component would dominate at
all energies. If we want both E_p\sim 300 keV and prompt HE emission up to 2
GeV, as detected by EGRET for GRB 940217, we need \Gamma\sim 600 and t_v\sim
0.1 ms, which might be resolved by super AGILE. If such prompt HE emission is
common in GRBs, as may be tested by GLAST, then for t_v\gtrsim 1 ms we need
\Gamma\gtrsim 350, which implies E_p\lesssim 100 keV. Therefore if X-ray
flashes are GRBs with high values of t_v and \Gamma, they should produce
\gtrsim 1 GeV emission. For an electron power law index p>2, the SSC component
dominates the emission above 100 MeV. Future observations by GLAST may help
determine the value of p and whether the HE emission is consistent with a
single power law (one component--the synchrotron, dominates) or has a break
where the \nuF_\nu slope turns from negative to positive, implying that the SSC
component becomes dominant above \sim 100 MeV. The HE emission is expected to
show similar variability and time structure to that of the soft gamma-ray
emission. Finally, we find that in order to see delayed HE emission from the
prompt GRB due to pair production with the cosmic IR background, extremely
small intergalactic magnetic fields (\lessim 10^{-22} G) are required.
