An explicit expression for the spectral distribution of the radiation energy from a relativistic electron with a magnetic moment is derived. Consequentially for an extreme relativistic electron moving along a planar circular orbit, the spectral distributions of synchrotron radiation in σ- and π- polarization are calculated including the modification from electron's magnetic moment. It is shown that for the synchrotron radiation with high critical frequency ωc with ħωc≥ 10 keV, the modification may be considerable. The relationship between the polarization of synchronous electrons and radiation fields is obtained, and thus the information about spin polarization of electron beam from measuring the polarization of radiation fields is provided.