The characteristic features of the resonant trident process (Oleinik resonances) have been theoretically studied in a wide range of frequencies and intensities of a circularly polarized strong electromagnetic wave. The resonant trident process is defined by two characteristic quantum energies: the characteristic energy of the nonlinear Compton effect and the characteristic energy of the nonlinear Breit-Wheeler process. These characteristic energies depend significantly on the frequency and intensity of the wave, as well as on the angle between the momenta of the initial electrons and the electromagnetic wave. The resonant trident process is effective when the energy of the initial electrons is greater than or on the order of magnitude of the corresponding characteristic energies. It is shown that quantum entanglement of final particles takes place in this resonant process. An important aspect of the resonant trident process is the equality of the energies of the electron and positron pairs. Analytical expressions for the differential rates of the resonant trident process on the energy of final particles are obtained. The corresponding analytical expressions for full rates have also been obtained. It is shown that the rate data of the resonant trident process in the field of optical and X-ray wave frequencies significantly exceed the corresponding rate of the non-resonant trident process. Results obtained can be used in experiments at leading laser centers, as well as to explain Quantum Electrodynamics (QED) processes in strong X-ray fields near neutron stars and magnetars.