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Abridged. A large fraction of stars are found in binary systems. It istherefore important for our understanding of the star formation process, toinvestigate the fragmentation of dense molecular cores. We study the influenceof the magnetic field, ideally coupled to the gas, on the fragmentation inmultiple systems of collapsing cores. We present high resolution numericalsimulations performed with the RAMSES MHD code starting with a uniform spherein solid body rotation and a uniform magnetic field parallel to the rotationaxis. We pay particular attention to the strength of the magnetic field andinterpret the results using the analysis presented in a companion paper. Theresults depend much on the amplitude, $A$, of the perturbations seededinitially. For a low amplitude, $A=0.1$, we find that for values of themass-to-flux over critical mass-to-flux ratio, $\mu$, as high as $\mu = 20$,the centrifugally supported disk which fragments in the hydrodynamical case, isstabilized and remains axisymmetric. Detailed investigations reveals that thisis due to the rapid growth of the toroidal magnetic field induced by thedifferential motions within the disk. For values of $\mu$ smaller $\simeq 5$,corresponding to larger magnetic intensities, there is no centrifugallysupported disk because of magnetic braking. When the amplitude of theperturbation is equal to $A=0.5$, each initial peak develops independently andthe core fragments for a large range of $\mu$. Only for values of $\mu$ closeto 1 is the magnetic field able to prevent the fragmentation. Since a largefraction of stars are binaries, the results of low magnetic intensitiespreventing the fragmentation in case of weak perturbations, is problematic. Wediscuss three possible mechanisms...

Magnetic processes in a collapsing dense core. II Fragmentation. Is there a fragmentation crisis ?