  Recent numerical studies suggest that magnetic fields play an important role
in primordial star formation in the early universe. However, the detailed
evolution of the magnetic field in the collapse phase still has uncertainties
because of the complicated physics associated with turbulence in a collapsing
magnetized system. Here, we perform a suite of numerical MHD simulations that
follow the collapse of magnetized, turbulent primordial gas clouds to
investigate the evolution of the magnetic field associated with the turbulence,
assuming a polytropic equation of state with exponent $\gamma_{\rm eff}$ and
with various numerical resolutions. In addition, we generalize the analytic
theory of magnetic field growth/saturation so that it can deal with various
exponents $\gamma_{\rm eff}$ and turbulence energy spectra. We find that the
numerical results are well reproduced by the theory for various $\gamma_{\rm
eff}$ through the collapse phase during the formation of the first stars. The
magnetic field is eventually amplified by a factor of $10^{12}$ -- $10^{15}$
due to kinematic and non-linear turbulent dynamo effects and reaches 3% -- 100%
of the equipartition level, depending on $\gamma_{\rm eff}$. We also find that
the transition between the kinematic and non-linear stages can be analytically
estimated. These results indicate that the strong magnetic field accompanied by
supersonic turbulence is a general property and suggest that it can play a
crucial role in the formation of the first stars.
