  We propose a simple renormalizable grand unified theory based on the $SU(5)$
gauge symmetry where the neutrino masses are generated at the quantum level
through the Zee mechanism. In this model the same Higgs needed to correct the
mass relation between charged leptons and down-type quarks plays a crucial role
to generate neutrino masses. We show that in this model one can satisfy the
constrains coming from the unification of gauge couplings and the mechanism for
neutrino masses is discussed in detail. The predictions for proton decay are
discussed in order to understand the testability at current and future
experiments such as Hyper-Kamiokande. This simple theory predicts a light
colored octet which could give rise to exotic signatures at the LHC.
