  Dijet production has been studied in neutral current deep inelastic e+p
scattering for 470 < Q**2 < 20000 GeV**2 with the ZEUS detector at HERA using
an integrated luminosity of 38.4 pb**{-1}. Dijet differential cross sections
are presented in a kinematic region where both theoretical and experimental
uncertainties are small. Next-to-leading-order (NLO) QCD calculations describe
the measured differential cross sections well. A QCD analysis of the measured
dijet fraction as a function of Q**2 allows both a precise determination of
alpha_s(M_Z) and a test of the energy-scale dependence of the strong coupling
constant. A detailed analysis provides an improved estimate of the
uncertainties of the NLO QCD cross sections arising from the parton
distribution functions of the proton. The value of alpha_s(M_Z), as determined
from the QCD fit, is alpha_s(M_Z) = 0.1166 +- 0.0019 (stat.) {+
0.0024}_{-0.0033} (exp.)} {+ 0.0057}_{- 0.0044} (th.).
