  Using polarization-resolved photoluminescence spectroscopy, we investigate
valley degeneracy breaking by out-of-plane magnetic field in back-gated
monolayer MoSe$_2$ devices. We observe a linear splitting of $-0.22
\frac{\text{meV}}{\text{T}}$ between luminescence peak energies in $\sigma_{+}$
and $\sigma_{-}$ emission for both neutral and charged excitons. The optical
selection rules of monolayer MoSe$_2$ couple photon handedness to the exciton
valley degree of freedom, so this splitting demonstrates valley degeneracy
breaking. In addition, we find that the luminescence handedness can be
controlled with magnetic field, to a degree that depends on the back-gate
voltage. An applied magnetic field therefore provides effective strategies for
control over the valley degree of freedom.
