  In the present-day universe, magnetic fields play such essential roles in
star formation as angular momentum transport and outflow driving, which control
circumstellar disc formation/fragmentation and also the star formation
efficiency. While only a much weaker field has been believed to exist in the
early universe, recent theoretical studies find that strong fields can be
generated by turbulent dynamo during the gravitational collapse. Here, we
investigate the gravitational collapse of a cloud core ($\sim 10^{3}\ \rm
cm^{-3}$) up to protostar formation ($\sim 10^{20}\ \rm cm^{-3}$) by non-ideal
magnetohydrodynamics (MHD) simulations considering ambipolar diffusion (AD),
the dominant non-ideal effects in the primordial-gas. We systematically study
rotating cloud cores either with or without turbulence and permeated with
uniform fields of different strengths. We find that AD can slightly suppress
the field growth by dynamo especially on scales smaller than the Jeans-scale at
the density range $10^{10}-10^{14}\ \rm cm^{-3}$, while we could not see the AD
effect on the temperature evolution, since the AD heating rate is always
smaller than compression heating. The inefficiency of AD makes the field as
strong as $10^{3}-10^{5} \rm\ G$ near the formed protostar, much stronger than
in the present-day cases, even in cases with initially weak fields. The
magnetic field affects the inflow motion when amplified to the equipartition
level with turbulence on the Jeans-scale, although disturbed fields do not
launch winds. This might suggest that dynamo amplified fields have smaller
impact on the dynamics in the later accretion phase than other processes such
as ionisation feedback.
