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Radiation spectrum of extreme relativistic electrons and a probability ofelectron-positron pair production by energetic photons in a strong plasmawakefield are derived in the framework of a semiclassical approach. It is shownthat that the radiation losses of the relativistic electron in the plasmawakefield scale as $ \propto \epsilon ^{2 / 3}$ in the quantum limit when theenergy of the radiated photon becomes close to the electron energy, $\epsilon$. The quantum effects will play a key role in future plasma-based acceleratorsoperating at ultrahigh energy of the electrons.Comment: 10 pages, 2 figure

Radiation emission by extreme relativistic electrons and pair production by hard photons in a strong plasma wakefield