Computer analysis of the field emission from multilayered cathodes of Si-SiO2-Si [quantum well (QW)]-SiO2 for different system parameters (the thicknesses of the SiO2 insulator and QW layers, carrier concentration in the substrate, crystal orientation, etc.) has shown that a well pronounced peak on the Fowler-Nordheim dependence appears. The origin of this peak and its location on the electric field axis is discussed. It is shown that they are caused by the resonance tunneling of electrons through the quantum levels in the quantum well.