  We derive general expressions at the one-loop level for the coefficients of
the covariant structure of the neutrino self-energy in the presence of a
constant magnetic field. The neutrino energy spectrum and index of refraction
are obtained for neutral and charged media in the strong-field limit ($M_{W}\gg
\sqrt{B}\gg m_{e},T,\mu ,| \mathbf{p}| $) using the lowest Landau level
approximation. The results found within the lowest Landau level approximation
are numerically validated, summing in all Landau levels, for strong $B\gg
T^{2}$ and weakly-strong $B \gtrsim T^{2}$ fields. The neutrino energy in
leading order of the Fermi coupling constant is expressed as the sum of three
terms: a kinetic-energy term, a term of interaction between the magnetic field
and an induced neutrino magnetic moment, and a rest-energy term. The leading
radiative correction to the kinetic-energy term depends linearly on the
magnetic field strength and is independent of the chemical potential. The other
two terms are only present in a charged medium. For strong and weakly-strong
fields, it is found that the field-dependent correction to the neutrino energy
in a neutral medium is much larger than the thermal one. Possible applications
to cosmology and astrophysics are considered.
