  The first galaxies form due to gravitational collapse of primordial halos.
During this collapse, weak magnetic seed fields get amplified exponentially by
the small-scale dynamo - a process converting kinetic energy from turbulence
into magnetic energy. We use the Kazantsev theory, which describes the
small-scale dynamo analytically, to study magnetic field amplification for
different turbulent velocity correlation functions. For incompressible
turbulence (Kolmogorov turbulence), we find that the growth rate is
proportional to the square root of the hydrodynamic Reynolds number, Re^(1/2).
In the case of highly compressible turbulence (Burgers turbulence) the growth
rate increases proportional to Re^(1/3). With a detailed chemical network we
are able to follow the chemical evolution and determine the kinetic and
magnetic viscosities (due to Ohmic and ambipolar diffusion) during the collapse
of the halo. This way, we can calculate the growth rate of the small-scale
dynamo quantitatively and predict the evolution of the small-scale magnetic
field. As the magnetic energy is transported to larger scales on the local
eddy-timescale, we obtain an estimate for the magnetic field on the Jeans
scale. Even there, we find that equipartition with the kinetic energy is
reached on small timescales. Dynamically relevant field structures can thus be
expected already during the formation of the first objects in the Universe.
