  The formation of Population III stars is investigated using resistive
magnetohydrodynamic simulations. Starting from a magnetized primordial
prestellar cloud, we calculate the cloud evolution several hundreds of years
after first protostar formation, resolving the protostellar radius. When the
natal minihalo field strength is weaker than B \lesssim 10^-13 (n/1 cm^-3)^-2/3
G (n is the hydrogen number density), magnetic effects can be ignored. In this
case, fragmentation occurs frequently and a stellar cluster forms, in which
stellar mergers and mass exchange between protostars contribute to the mass
growth of these protostars. During the early gas accretion phase, the most
massive protostar remains near the cloud centre, whereas some of the less
massive protostars are ejected. The magnetic field significantly affects
Population III star formation when B_amb \gtrsim 10^-12 (n/1 cm^-3)^-2/3 G. In
this case, because the angular momentum around the protostar is effectively
transferred by both magnetic braking and protostellar jets, the gas falls
directly onto the protostar without forming a disk, and only a single massive
star forms. In addition, a massive binary stellar system appears when B_amb
\sim 10^-12 (n/1cm^-3)^-2/3 G. Therefore, the magnetic field determines the end
result of the formation process (cluster, binary or single star) for Population
III stars. Moreover, no persistent circumstellar disk appears around the
protostar regardless of the magnetic field strength, which may influence the
further evolution of Population III stars.
