  The energy relaxation channels of hot electrons far from thermal equilibrium
in a degenerate two-dimensional electron system are investigated in transport
experiments in a mesoscopic three-terminal device. We observe a transition from
two dimensions at zero magnetic field to quasi--one-dimensional scattering of
the hot electrons in a strong magnetic field. In the two-dimensional case
electron-electron scattering is the dominant relaxation mechanism, while the
emission of optical phonons becomes more and more important as the magnetic
field is increased. The observation of up to 11 optical phonons emitted per hot
electron allows us to determine the onset energy of LO phonons in GaAs at
cryogenic temperatures with a high precision, $\eph=36.0\pm0.1\,$meV. Numerical
calculations of electron-electron scattering and the emission of optical
phonons underline our interpretation in terms of a transition to
one-dimensional dynamics.
