  We provide a theoretical description of synchrotron fluctuations arising from
magnetic turbulence. We derive an expression that relates the correlation of
synchrotron fluctuations for an arbitrary index of relativistic electrons with
the correlations of squared fluctuations of magnetic field component
perpendicular to the line of sight. The latter correlations we study assuming
that the turbulence is axisymmetric. We obtain general relations valid for an
arbitrary model of magnetic turbulence and analyse the relations for particular
example of magnetic turbulence that is supported by numerical simulations. We
predict that the synchrotron intensity fluctuations are anisotropic with larger
correlation present along the direction of magnetic field. This anisotropy is
dominated by the quadrupole component with the ratio between quadrupole and
monopole parts being sensitive to the compressibility of underlying turbulence.
Our work opens avenues for quantitative studies of magnetic turbulence in our
galaxy and beyond using synchrotron emission. It also outlines the directions
of how synchrotron foreground emission can be separated from cosmological
signal, i.e. from CMB or highly redshifted HI emission. For the sake of
completeness we also provide the expressions for the synchrotron polarization
(Stocks parameters and their combinations) for the model of axisymmetric
magnetic turbulence.
