The spectrum of the two-dimensional continuum Dirac operator
in the presence of a uniform background magnetic field consists
of Landau levels, which are degenerate and separated by gaps. On
the lattice the Landau levels are spread out by discretization
artefacts, but a remnant of their structure is clearly visible
(Hofstadter butterfly). If one switches on a non-Abelian interaction,
the butterfly structure will be smeared out, but the lowest Landau
level (LLL) will still be separated by a gap from the rest of the spectrum. In this paper we discuss how one can define the LLL in QCD and check how well certain physical quantities are approximated by taking into account
only the LLL.