  The inverse Compton (IC) interpretation of the excess EUV emission, that was
recently reported from several clusters of galaxies, suggests that the amount
of relativistic electrons in the intracluster medium is highly significant,
W_e>10^{61} erg. Considering Coma as the prototype galaxy cluster of nonthermal
radiation, we discuss implications of the inverse Compton origin of the excess
EUV fluxes in the case of low intracluster magnetic fields of order 0.1 muG, as
required for the IC interpretation of the observed excess hard X-ray flux, and
in the case of high fields of order 1 muG as suggested by Faraday rotation
measurements. Although for such high intracluster fields the excess hard X-rays
will require an explanation other than by the IC effect, we show that the
excess EUV flux can be explained by the IC emission of a `relic' population of
electrons driven into the incipient intracluster medium at the epoch of
starburst activity by galactic winds, and later on reenergized by adiabatic
compression and/or large-scale shocks transmitted through the cluster as the
consequence of more recent merger events. For high magnetic fields B > 1 muG
the interpretation of the radio fluxes of Coma requires a second population of
electrons injected recently. They can be explained as secondaries produced by a
population of relativistic protons. We calculate the fluxes of gamma-rays to be
expected in both the low and high magnetic field scenarios, and discuss
possibilities to distinguish between these two principal options by future
gamma-ray observations.
