  While present standard model of cosmology yields no clear prediction for the
initial magnetic field strength, efficient dynamo action may compensate for
initially weak seed fields via rapid amplification. In particular, the
small-scale dynamo is expected to exponentially amplify any weak magnetic field
in the presence of turbulence. We explore whether this scenario is viable using
cosmological magneto-hydrodynamics simulations modeling the formation of the
first galaxies, which are expected to form in so-called atomic cooling halos
with virial temperatures $\rm T_{vir} \geq 10^{4}$ K. As previous calculations
have shown that a high Jeans resolution is needed to resolve turbulent
structures and dynamo effects, our calculations employ resolutions of up to 128
cells per Jeans length. The presence of the dynamo can be clearly confirmed for
resolutions of at least 64 cells per Jeans length, while saturation occurs at
approximate equipartition with turbulent energy. As a result of the large
Reynolds numbers in primordial galaxies, we expect saturation to occur at early
stages, implying magnetic field strengths of \sim0.1 $\mu$G at densities of
10^4 cm^{-3}.
