  We present for the first time a self-contained calculation of the QCD running
coupling at finite temperature and quark chemical potential, alpha_s(T, mu),
based on a semiclassical background field method. The hard thermal/dense loop
results on the Debye screening mass are recovered in a first approximation. The
final result can be interpreted as the ordinary zero temperature running
coupling, with momenta replaced by in-medium scales Lambda: at high density and
zero temperature, the quark scale is set by Lambda ~ 24.4 mu. At high
temperature and moderate densities, the quark in-medium scale reads Lambda^2 ~
[2.91 T]^2 + [1.91 mu]^2, which resembles the naive phenomenological estimate
Lambda^2 = (pi T)^2 + mu^2.
