  Recent FERMI observations provide a lower limit of 10^{-15} G for the
magnetic field strength in the intergalactic medium (IGM). This is consistent
with theoretical expectations based on the Biermann battery effect, which
predicts such IGM fields already at redshifts z~10. During gravitational
collapse, such magnetic fields can be amplified by compression and by
turbulence, giving rise to the small-scale dynamo. On scales below the Jeans
length, the eddy turnover timescale is much shorter than the free-fall
timescale, so that saturation can be reached during collapse. This scenario has
been tested and confirmed with magneto-hydrodynamical simulations following the
collapse of a turbulent, weakly magnetized cloud. Based on a spectral analysis,
we confirm that turbulence is injected on the Jeans scale. For the power
spectrum of the magnetic field, we obtain the Kazantsev slope which is
characteristic for the small-scale dynamo. A calculation of the critical length
scales for ambipolar diffusion and Ohmic dissipation shows that these scales
are always small enough to allow significant amplification of the magnetic
field by small-scale eddies. We discuss potential implications for the
protostellar accretion disk, with particular focus on the magneto-rotational
instability, which may change the morphology of the disk and reduce the
accretion rate by a factor of a few.
