  We study the synchrotron emission of neutrino pairs by relativistic,
degenerate electrons in strong magnetic fields. Particular attention is paid to
the case in which the dominant contribution comes from one or several lowest
cyclotron harmonics. Calculations are performed using the exact quantum
formalism and the quasiclassical approach. Simple analytic fits to the neutrino
synchrotron emissivity are obtained in the domain of magnetized, degenerate and
relativistic electron gas provided the electrons populate either many Landau
levels or the ground level alone. The significance of the neutrino synchrotron
energy losses in the interiors of cooling neutron stars is discussed.
