[en] The subject of this investigation is photon radiation by an electron in a homogeneous magnetic field Hvector and in the field of a circularly polarized electromagnetic wave (EMW) propagating along Hvector. The electron-external field interaction is taken into account by using correct solutions to the Dirac equation, and the electron-quantized electromagnetic field interaction, in the first order of the perturbation theory. A quasi-energy spectrum of an electron is obtained which depends essentially on EMW polarization. In the case of elliptic polarization the spectrum consists of four energy zones and in the case of circular polarization - of three energy zones. The origin of the complex picture of an electron zone spectrum is elucidated. The energy spectrum of photons emitted by an electron in the field of a circularly polarized EMW is investigated. It is shown that the frequency shift Δω' of a photon which depends on the EMW intensity I, is the greatest if initial state electrons have pulses which correspond to the intersection point of dispersion curves in the quasi-energy spectrum (in this case Δω'approximately √1). An estimate of the scattering probability is also given