Context :

Aims : 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 .

Methods : The MHD equations are solved in a simplified model with cylindric geometry .
We assume the axial field as uniform and the fluid as incompressible .

Results : 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 Alfvén 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 .

Conclusions : 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 \leq 1 .
The corresponding growth rates are of the order of the rotation period of the inner part of the galaxy .