  Fermi observations of high-energy gamma-ray emission from GRB 080916C shows
that its spectrum is consistent with the Band function from MeV to tens of GeV.
Assuming one single emission mechanism dominates in the whole energy range, we
show that this spectrum is consistent with synchrotron origin by
shock-accelerated electrons. The simple electron inverse-Compton model and the
hadronic model are found to be less viable. In the synchrotron scenario, the
synchrotron self-Compton scattering is likely to be in the Klein-Nishina regime
and therefore the resulting high-energy emission is subdominant, even though
the magnetic field energy density is lower than that in relativistic electrons.
The Klein-Nishina inverse-Compton cooling may also affect the low-energy
electron number distribution and hence results in a low-energy synchrotron
photon spectrum $n(\nu)\propto\nu^{-1}$ below the peak energy. Under the
framework of the electron synchrotron interpretation, we constrain the shock
microphysical parameters and derive a lower limit of the upstream magnetic
fields. The detection of synchrotron emission extending to about 70 GeV in the
source frame in GRB 080916C favors the Bohm diffusive shock acceleration if the
bulk Lorentz factor of the relativistic outflow is not significantly greater
than thousands.
