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The gain of an antenna is the ratio of the maxiumum field strength produced by the antenna at a given distance compared to a 100% efficient reference antenna with the same power input. The gain is the directivity of the antenna multiplied by its efficiency. If the efficiency is 1, or 100%, the gain is the same as the directivity.
The 3 dB difference noted in the previous two posts (which, as Rich correctly says, results from the fact that all of the radiation from the 1/4 wave monopole over a perfect ground plane is confined to one hemisphere) is the difference in the gains of the two types of antennas. Because of the 3 dB difference, the gains cannot be the same. The 3 dB difference is a difference in gain of 3 dB. It does not matter what the reason for this 3 dB difference is for there to be a gain difference.
The reference antenna is usually accepted to be the isotropic radiator, which is postulated to have completely uniform radiation in all space directions. Its gain is 0 dBi, by definition. The field strength at 1 km for 1 kW of radiated power is about 173.2 mV/m.
The half-wave dipole has a particular doughnut-shaped radiation pattern and has a gain of about 2.15 dBi. The field strength at 1 km for 1 kW of radiated power is about 221.5 mV/m.
The quarter-wave monopole above a perfect ground plane has a radiation pattern shaped the same as for the half-wave dipole, except that the dougnut-shape is cut in half. Its gain is about 5.15 dBi. The field strength at 1 km for 1 kW of radiated power is about 313.2 mV/m
The editions of Kraus (1950) and Jasik (1961) that I have both give the definition of gain I give here.