  We theoretically study the influence of laser polarization in inelastic
scattering of electrons by hydrogen atoms in the presence of a circularly
polarized laser field in the domain of field strengths below $10^{7}$ V/cm and
high projectile energies. A semi-perturbative approach is used in which the
interaction of the projectile electrons with the laser field is described by
Gordon-Volkov wave functions, while the interaction of the hydrogen atom with
the laser field is described by first-order time-dependent perturbation theory.
A \textit{closed analytical solution} is derived in laser-assisted inelastic
electron-hydrogen scattering for the $1s \to nl$ excitation cross section which
is valid for both circular and linear polarizations. For the excitation of the
$n=2$ levels simple analytical expressions of differential cross section are
derived for laser-assisted inelastic scattering in the perturbative domain, and
the differential cross sections by the circularly and linearly polarized laser
fields and their ratios for one- and two-photon absorption are calculated as a
function of the scattering angle. Detailed numerical results for the angular
dependence and the resonance structure of the differential cross sections is
discussed for the $1s \to 4l$ excitations of hydrogen in a circularly polarized
laser field.
