[en] One of the most promising approaches to advanced propulsion that could meet the objectives of the Space Exploration Initiative (SEI) is the open cycle gas core nuclear rocket (GCR). The energy in this device is generated by a fissioning uranium plasma which heats, through radiation, a propellant that flows around the core and exits through a nozzle, thereby converting thermal energy into thrust. Although such a scheme can produce very attractive propulsion parameters in the form of high specific impulse and high thrust, it does suffer from serious physics and engineering problems that must be addressed if it is to become a viable propulsion system. Among the major problems that must be solved are the confinement of the uranium plasma, potential instabilities and control problems associated with the dynamics of the uranium core, and the question of startup and fueling of such a reactor. In this paper, the authors focus their attention on the problems of equilibria and stability of the uranium care, and examine the potential use of an externally applied magnetic field for these purposes. They find that steady state operation of the reactor is possible only for certain care profiles that may not be compatible with the radiative aspect of the system. The authors also find that the system is susceptible to hydrodynamic and acoustic instabilities that could deplete the uranium fuel in a short time if not properly suppressed