  Inclusive jet, dijet and trijet differential cross sections are measured in
neutral current deep-inelastic scattering for exchanged boson virtualities 150
< Q^2 < 15000 GeV^2 using the H1 detector at HERA. The data were taken in the
years 2003 to 2007 and correspond to an integrated luminosity of 351 pb^{-1}.
Double differential Jet cross sections are obtained using a regularised
unfolding procedure. They are presented as a function of Q^2 and the transverse
momentum of the jet, P_T^jet, and as a function of Q^2 and the proton's
longitudinal momentum fraction, Xi, carried by the parton participating in the
hard interaction. In addition normalised double differential jet cross sections
are measured as the ratio of the jet cross sections to the inclusive neutral
current cross sections in the respective Q^2 bins of the jet measurements.
Compared to earlier work, the measurements benefit from an improved
reconstruction and calibration of the hadronic final state. The cross sections
are compared to perturbative QCD calculations in next-to-leading order and are
used to determine the running coupling and the value of the strong coupling
constant as alpha_s(M_Z) = 0.1165 (8)_exp (38)_{pdf,theo}.
