  We analyze the running gauge coupling at finite temperature for QCD, using
the functional renormalization group. The running of the coupling is calculated
for all scales and temperatures. At finite temperature, the coupling is
governed by a fixed point of the 3-dimensional theory for scales smaller than
the corresponding temperature. The running coupling can drive the quark sector
to criticality, resulting in chiral symmetry breaking. Our results provide for
a quantitative determination of the phase boundary in the plane of temperature
and number of massless flavors. Using the experimental value of the coupling at
the tau mass scale as the only input parameter, we obtain, e.g., for N_f=3
massless flavors a critical temperature of T_cr ~ 148 MeV in good agreement
with lattice simulations.
