  A new measurement of inclusive-jet cross sections in the Breit frame in
neutral current deep inelastic scattering using the ZEUS detector at the HERA
collider is presented. The data were taken in the years 2004 to 2007 at a
centre-of-mass energy of $318\,\text{GeV}$ and correspond to an integrated
luminosity of $347\,\text{pb}^{-1}$. Massless jets, reconstructed using the
$k_t$-algorithm in the Breit reference frame, have been measured as a function
of the squared momentum transfer, $Q^2$, and the transverse momentum of the
jets in the Breit frame, $p_{\perp,\text{Breit}}$. The measured jet cross
sections are compared to previous measurements and to perturbative QCD
predictions. The measurement has been used in a next-to-next-to-leading-order
QCD analysis to perform a simultaneous determination of parton distribution
functions of the proton and the strong coupling, resulting in a value of
$\alpha_s(M_Z^2) = 0.1142 \pm 0.0017~\text{(experimental/fit)}$
${}^{+0.0006}_{-0.0007}~\text{(model/parameterisation)}$
${}^{+0.0006}_{-0.0004}~\text{(scale)}$, whose accuracy is improved compared to
similar measurements. In addition, the running of the strong coupling is
demonstrated using data obtained at different scales.
