  To find out whether toroidal field can stably exist in galaxies the
current-driven instability of toroidal magnetic fields is considered under the
influence of an axial magnetic field component and under the influence of both
rigid and differential rotation. The MHD equations are solved in a simplified
model with cylindric geometry. We assume the axial field as uniform and the
fluid as incompressible. The stability of a toroidal magnetic field is strongly
influenced by uniform axial magnetic fields. If both field components are of
the same order of magnitude then the instability is slightly supported and
modes with m>1 dominate. If the axial field even dominates the most unstable
modes have again m>1 but the field is strongly stabilized. All modes are
suppressed by a fast rigid rotation where the m=1 mode maximally resists. Just
this mode becomes best re-animated for \Omega > \Omega^A (\Omega^A the Alfven
frequency) if the rotation has a negative shear. -- Strong indication has been
found for a stabilization of the nonaxisymmetric modes for fluids with small
magnetic Prandtl number if they are unstable for Pm=1. For rotating fluids the
higher modes with m>1 do not play an important role in the linear theory. In
the light of our results galactic fields should be marginally unstable against
perturbations with m<= 1. The corresponding growth rates are of the order of
the rotation period of the inner part of the galaxy.
