  Itinerant ferromagnetism, i.e. spontaneous polarization of non-localized
particles, is expected to occur for strong repulsive interactions in a spin-1/2
Fermi system. However, this state has proven notoriously hard to find
experimentally, both in ultracold gases and in solids. This raises questions
about the stability of the itinerant ferromagnetic state itself. Here we
develop a new approach to describe both the direct and exchange interactions
for a general interaction potential in the path-integral formalism and we apply
this method to itinerant ferromagnetism in three-dimensional ultracold Fermi
gases. We show that the exchange interactions are lost in the
Hubbard-Stratonovich transformation and we propose to explicitly include the
exchange effects in a new modified interaction potential. In the saddle-point
approximation, the effect of interactions can be taken into account using only
three parameters. If the interactions become too strong, all saddle points
become unstable to density fluctuations. This greatly restricts the area in the
phase diagram where uniform itinerant ferromagnetism is expected to occur.
