  Inclusive jet differential cross sections have been measured in neutral
current deep inelastic e+p scattering for boson virtualities Q**2>125 GeV**2.
The data were taken using the ZEUS detector at HERA and correspond to an
integrated luminosity of 38.6 pb-1. Jets were identified in the Breit frame
using the longitudinally invariant K_T cluster algorithm. Measurements of
differential inclusive jet cross sections are presented as functions of jet
transverse energy (E_T,jet), jet pseudorapidity and Q**2, for jets with
E_T,jet>8 GeV. Next-to-leading-order QCD calculations agree well with the
measurements both at high Q**2 and high E_T,jet. The value of alpha_s(M_Z),
determined from an analysis of dsigma/dQ**2 for Q**2>500 GeV**2, is
alpha_s(M_Z) = 0.1212 +/- 0.0017 (stat.) +0.0023 / -0.0031 (syst.) +0.0028 /
-0.0027 (th.).
