  We compute for the first time the complete corrections at N$^3$LO in the
strong coupling constant to the inclusive neutral-current Drell-Yan process
including contributions from both photon and $Z$-boson exchange. Our main
result is the computation of the QCD corrections to the inclusive production
cross section of an axial-vector boson to third order in the strong coupling in
a variant of QCD with five massless quark flavours. Since the axial anomaly
does not cancel for an odd number of flavours, we also consistently include
non-decoupling effects in the top-quark mass through three loops. We perform a
phenomenological study of our results, and we present for the first time
predictions for the inclusive Drell-Yan process at the LHC at this order in QCD
perturbation theory.
