  We study the buildup of magnetic fields during the formation of Population
III star-forming regions, by conducting cosmological simulations from realistic
initial conditions and varying the Jeans resolution. To investigate this in
detail, we start simulations from identical initial conditions, mandating 16,
32 and 64 zones per Jeans length, and studied the variation in their magnetic
field amplification. We find that, while compression results in some
amplification, turbulent velocity fluctuations driven by the collapse can
further amplify an initially weak seed field via dynamo action, provided there
is sufficient numerical resolution to capture vortical motions (we find this
requirement to be 64 zones per Jeans length, slightly larger than, but
consistent with previous work run with more idealized collapse scenarios). We
explore saturation of amplification of the magnetic field, which could
potentially become dynamically important in subsequent, fully-resolved
calculations. We have also identified a relatively surprising phenomena that is
purely hydrodynamic: the higher-resolved simulations possess substantially
different characteristics, including higher infall-velocity, increased
temperatures inside 1000 AU, and decreased molecular hydrogen content in the
innermost region. Furthermore, we find that disk formation is suppressed in
higher-resolution calculations, at least at the times that we can follow the
calculation. We discuss the effect this may have on the buildup of disks over
the accretion history of the first clump to form as well as the potential for
gravitational instabilities to develop and induce fragmentation.
