  We study the quantum properties of a Galilean-invariant abelian gauge theory
coupled to a Schr\"odinger scalar in 2+1 dimensions. At the classical level,
the theory with minimal coupling is obtained from a null-reduction of
relativistic Maxwell theory coupled to a complex scalar field in 3+1 dimensions
and is closely related to the Galilean electromagnetism of Le-Bellac and
L\'evy-Leblond. Due to the presence of a dimensionless, gauge-invariant scalar
field in the Galilean multiplet of the gauge-field, we find that at the quantum
level an infinite number of couplings is generated. We explain how to handle
the quantum corrections systematically using the background field method. Due
to a non-renormalization theorem, the beta function of the gauge coupling is
found to vanish to all orders in perturbation theory, leading to a continuous
family of fixed points where the non-relativistic conformal symmetry is
preserved.
