  We study the local stability of stratified, differentially-rotating fluids to
axisymmetric perturbations in the presence of a weak magnetic field and of
finite resistivity, viscosity and heat conductivity. This is a generalization
of the Goldreich-Schubert-Fricke (GSF) double-diffusive analysis to the
magnetized and resistive, triple-diffusive case. Our fifth-order dispersion
relation admits a novel branch which describes a magnetized version of
multi-diffusive modes. We derive necessary conditions for axisymmetric
stability in the inviscid and perfect-conductor (double-diffusive) limits. In
each case, rotation must be constant on cylinders and angular velocity must not
decrease with distance from the rotation axis for stability, irrespective of
the relative strength of viscous, resistive and heat diffusion. Therefore, in
both double-diffusive limits, solid body rotation marginally satisfies our
stability criteria. The role of weak magnetic fields is essential to reach
these conclusions. The triple-diffusive situation is more complex, and its
stability criteria are not easily stated. Numerical analysis of our general
dispersion relation confirms our analytic double-diffusive criteria, but also
shows that an unstable double-diffusive situation can be significantly
stabilized by the addition of a third, ostensibly weaker, diffusion process. We
describe a numerical application to the Sun's upper radiative zone and
establish that it would be subject to unstable multi-diffusive modes if
moderate or strong radial gradients of angular velocity were present.
