  We compute the differential cross-section for inclusive prompt photon
production in deeply inelastic scattering (DIS) of electrons on nuclei at small
$x$ in the framework of the Color Glass Condensate (CGC) effective theory. The
leading order (LO) computation in this framework resums leading logarithms in
$x$ as well as power corrections to all orders in $Q_{s,A}^2/Q^2$, where
$Q_{s,A}(x)$ is the nuclear saturation scale. This LO result is proportional to
universal dipole and quadrupole Wilson line correlators in the nucleus. In the
soft photon limit, the Low-Burnett-Kroll theorem allows us to recover existing
results on inclusive DIS dijet production. The $k_{\perp}$ and collinearly
factorized expressions for prompt photon production in DIS are also recovered
in a leading twist approximation to our result. In the latter case, our result
corresponds to the dominant next-to-leading order (NLO) perturbative QCD
contribution at small $x$. We next discuss the computation of the NLO
corrections to inclusive prompt photon production in the CGC framework. In
particular, we emphasize the advantages for higher order computations in
inclusive photon production, and for fully inclusive DIS, arising from the
simple momentum space structure of the dressed quark and gluon "shock wave"
propagators in the "wrong" light cone gauge $A^-=0$ for a nucleus moving with
$P^{+}_{N} \rightarrow \infty$.
