  We examine the effect of a parasitic rf magnetic field, attributed to ion
trapping, on the highly anticipated nuclear clock based on $^{229}$Th$^{3+}$
[C. J. Campbell et al., Phys. Rev. Lett. 108, 120802 (2012)]. The rf magnetic
field induces an ac Zeeman shift to the clock frequency. As we demonstrate,
this shift threatens to be the dominant systematic frequency shift for the
clock, exceeding other systematic frequency shifts and the projected systematic
uncertainty of the clock by orders of magnitude. We propose practical means to
suppress or eliminate this shift.
