  Are the kG-strength magnetic fields observed in young stars a fossil field
left over from their formation or are they generated by a dynamo? Our previous
numerical study concluded that magnetic fields must originate by a dynamo
process. Here, we continue that investigation by performing even higher
numerical resolution calculations of the gravitational collapse of a
1~M$_\odot$ rotating, magnetised molecular cloud core through the first and
second collapse phases until stellar densities are reached. Each model includes
Ohmic resistivity, ambipolar diffusion, and the Hall effect. We test six
numerical resolutions, using between $10^5$ and $3\times10^7$ particles to
model the cloud. At all but the lowest resolutions, magnetic walls form in the
outer parts of the first hydrostatic core, with the maximum magnetic field
strength located within the wall rather than at the centre of the core. At high
resolution, this magnetic wall is disrupted by the Hall effect, producing a
magnetic field with a spiral-shaped distribution of intensity. As the second
collapse occurs, this field is dragged inward and grows in strength, with the
maximum field strength increasing with resolution. As the second core forms,
the maximum field strength exceeds 1~kG in our highest resolution simulations,
and the stellar core field strength exceeds this threshold at the highest
resolution. Our resolution study suggests that kG-strength magnetic fields may
be implanted in low-mass stars during their formation, and may persist over
long timescales given that the diffusion timescale for the magnetic field
exceeds the age of the Universe.
