  We study interacting electrons in a periodic potential and a uniform magnetic
field ${\bf B}$ taking the spin-orbit interaction into account. We first
establish a perturbation expansion for those electrons with respect to the
Bloch states in zero field. It is shown that the expansion can be performed
with the zero-field Feynman diagrams of satisfying the momentum and energy
conservation laws. We thereby clarify the structures of the self-energy and the
thermodynamic potential in a finite magnetic field. We also provide a
prescription of calculating the electronic structure in a finite magnetic field
within the density functional theory starting from the zero-field energy-band
structure. On the basis of these formulations, we derive explicit expressions
for the magnetic susceptibility of ${\bf B}\to{\bf 0}$ at various approximation
levels on the interaction, particularly within the density functional theory,
which include the result of Roth [J. Phys. Chem. Solids {\bf 23} (1962) 433] as
the non-interacting limit. We finally study the de Haas-van Alphen oscillation
in metals to show that quasiparticles at the Fermi level with the many-body
effective mass are directly relevant to the phenomenon. The present argument
may be more transparent than that by Luttinger [Phys. Rev. {\bf 121} (1961)
1251] of using the gauge invariance and has an advantage that the change of the
band structure with the field may be incorporated.
