We investigate the formation and evolution of a first core , protostar , and circumstellar disc with a three-dimensional non-ideal ( including both Ohmic and ambipolar diffusion ) radiation magnetohydrodynamics simulation .
We found that the magnetic flux is largely removed by magnetic diffusion in the first core phase and that the plasma \beta of the centre of the first core becomes large , \beta > 10 ^ { 4 } .
Thus , proper treatment of first core phase is crucial in investigating the formation of protostar and disc .
On the other hand , in an ideal simulation , \beta \sim 10 at the centre of the first core .
The simulations with magnetic diffusion show that the circumstellar disc forms at almost the same time of protostar formation even with a relatively strong initial magnetic field ( the value for the initial mass-to-flux ratio of the cloud core relative to the critical value is \mu = 4 ) .
The disc has a radius of r \sim 1 AU at the protostar formation epoch .
We confirm that the disc is rotationally supported .
We also show that the disc is massive ( Q \sim 1 ) and that gravitational instability may play an important role in the subsequent disc evolution .