  We derive finite temperature expansions for relativistic fermion systems in
the presence of background magnetic fields, and with nonzero chemical
potential. We use the imaginary-time formalism for the finite temperature
effects, the proper-time method for the background field effects, and zeta
function regularization for developing the expansions. We emphasize the
essential difference between even and odd dimensions, focusing on $2+1$ and
$3+1$ dimensions. We concentrate on the high temperature limit, but we also
discuss the $T=0$ limit with nonzero chemical potential.
