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Theoretical and numerical works indicate that a strong magnetic field should suppress fragmentation in dense cores. However, this has never been tested observationally in a relatively large sample of fragmenting massive dense cores. Here we use the polarization data obtained in the Submillimeter Array Legacy Survey of Zhang et al. to build a sample of 18 massive dense cores where both fragmentation and magnetic field properties are studied in a uniform way. We measured the fragmentation level, Nmm, within the field of view common to all regions, of 0.15 pc, with a mass sensitivity of about 0.5 Msun, and a spatial resolution of about 1000 au. In order to obtain the magnetic field strength using the Davis-Chandrasekhar-Fermi method, we estimated the dispersion of the polarization position angles, the velocity dispersion of the H13CO+(4-3) gas, and the density of each core, all averaged within 0.15 pc. A strong correlation is found between Nmm and the average density of the parental core, although with significant scatter. When large-scale systematic motions are separated from the velocity dispersion and only the small-scale (turbulent) contribution is taken into account, a tentative correlation is found between Nmm and the mass-to-flux ratio, as suggested by numerical and theoretical works.