  We quantize to completion an inflationary universe with small inhomogeneities
in the framework of loop quantum cosmology. The homogeneous setting consists of
a massive scalar field propagating in a closed, homogeneous scenario. We
provide a complete quantum description of the system employing loop
quantization techniques. After introducing small inhomogeneities as scalar
perturbations, we identify the true physical degrees of freedom by means of a
partial gauge fixing, removing all the local degrees of freedom except the
matter perturbations. We finally combine a Fock description for the
inhomogeneities with the polymeric quantization of the homogeneous background,
providing the quantum Hamiltonian constraint of the composed system. Its
solutions are then completely characterized, owing to the suitable choice of
quantum constraint, and the physical Hilbert space is constructed. Finally, we
consider the analog description for an alternate gauge and, moreover, in terms
of gauge-invariant quantities. In the deparametrized model, all these
descriptions are unitarily equivalent at the quantum level.
