  The exchange interaction between two magnetic moments embedded in a host
metal is fundamental to the description of the magnetic behavior of solids. In
the standard spin-degenerate electron gas, it leads to the well known
Ruderman-Kittel-Kasuya-Yoshida (RKKY) interaction, which is of the Heisenberg
form $J\vec S_1\cdot \vec S_2$, where the interaction strength $J$ oscillates
and falls off with the distance between the moments in a characteristic way.
Here, we study the more general case of the spin-polarized electron gas both in
two and three dimensions, by evaluating the interaction strength as an
integration over the product of the host Green's functions. We find that in
addition to the Heisenberg term, an additional Ising-like term appears in the
magnetic interaction, so that the net interaction for the spin-polarized gas is
of the form $J_1 \vec S_1\cdot \vec S_2 + J_2 S_{1z} S_{2z}$. The interactions
show a beating pattern as a function of distance, caused by the two different
Fermi momenta for the two spins.
