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We discuss the bosonization of a relativistic very dense Fermi gas in a magnetic field and the consequent BoseEinstein condensation of the resulting relativistic vector gas of charged particles. The model may be applied to paired spin-up electrons. We show that such systems may maintain self-consistently magnetic fields of order 1010 - 1013 G. That pairing could be the origin of large magnetic fields in some white dwarfs and neutron stars.For fields large enough (～ 1013 for white dwarfs), the system becomes unstable and collapses.