  Here we summarize our recent results of high-resolution computer simulations
on the turbulent amplification of weak magnetic seed fields showing that such
fields will be exponentially amplified also during the gravitational collapse
reminiscent to the situation during primordial star formation. The exponential
magnetic field amplification is driven by the turbulent small-scale dynamo that
can be only observed in computer simulations if the turbulent motions in the
central core are sufficiently resolved. We find that the Jeans length, which
determines the central core region, has to be resolved by at least 30 grid
cells to capture the dynamo activity. We conclude from our studies that strong
magnetic fields will be unavoidably created already during the formation of the
first stars in the Universe, potentially influencing their evolution and mass
distribution.
