  We study the amplification of magnetic fields during the formation of
primordial halos. The turbulence generated by gravitational infall motions
during the formation of the first stars and galaxies can amplify magnetic
fields very efficiently and on short timescales up to dynamically significant
values. Using the Kazantsev theory, which describes the so-called small-scale
dynamo - a magnetohydrodynamical process converting kinetic energy from
turbulence into magnetic energy - we can then calculate the growth rate of the
small-scale magnetic field. Our calculations are based on a detailed chemical
network and we include non-ideal magnetohydrodynamical effects such as
ambipolar diffusion and Ohmic dissipation. We follow the evolution of the
magnetic field up to larger scales until saturation occurs on the Jeans scale.
Assuming a weak magnetic seed field generated by the Biermann battery process,
both Burgers and Kolmogorov turbulence lead to saturation within a rather small
density range. Such fields are likely to become relevant after the formation of
a protostellar disk and, thus, could influence the formation of the first stars
and galaxies in the Universe.
