  Polarization-resolved magneto-luminescence, together with simultaneous
magneto-transport measurements, have been performed on a two-dimensional
electron gas (2DEG) confined in CdTe quantum well in order to determine the
spin-splitting of fully occupied electronic Landau levels, as a function of the
magnetic field (arbitrary Landau level filling factors) and temperature. The
spin splitting, extracted from the energy separation of the \sigma+ and \sigma-
transitions, is composed of the ordinary Zeeman term and a many-body
contribution which is shown to be driven by the spin-polarization of the 2DEG.
It is argued that both these contributions result in a simple, rigid shift of
Landau level ladders with opposite spins.
