  The properties of a two-dimensional electron gas (2DEG) in a semiconductor
host with two valleys related by an underlying $C_4$ rotational symmetry are
studied using Hartree-Fock (HF) and various other many-body approaches. A
familiar artifact of the HF approach is a degeneracy between the valley
polarized - ``Ising nematic'' - and spin polarized - ferromagnetic - phases,
which is inconsistent with recent variational Monte Carlo (VMC) results.
Correlation effects, computed either within the random phase approximation
(RPA) or the T-matrix approximation, enhance the valley susceptibility relative
to the spin susceptibility. Extrapolating the results to finite interaction
strength, we find a direct first-order transition from a symmetry-unbroken
state to a spin unpolarized Ising nematic fluid with full valley polarization,
in qualitative agreement with VMC. The RPA results are also reminiscent of
experiments on the corresponding 2DEG in AlAs heterostructures.
