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[en] The effect of the electron's anomalous magnetic moment on the relativistic electronic dressing for the process of electron-hydrogen atom elastic collisions is investigated. We consider a laser field with circular polarization and various electric field strengths. The Dirac-Volkov states taking into account this anomaly are used to describe the process in the first order of perturbation theory. The correlation between the terms coming from this anomaly and the electric field strength gives rise to the strong dependence of the spinor part of the differential cross section (DCS) with respect to these terms. A detailed study has been devoted to the nonrelativistic regime as well as the moderate relativistic regime. Some aspects of this dependence as well as the dynamical behavior of the DCS in the relativistic regime have been addressed.