Jet Production at Low and High Q² and Determination of the Strong Coupling α_s at H1

Two recent measurements of inclusive jet, 2-jet and 3-jet cross sections in deep-inelastic $ep$ scattering from the H1 collaboration are presented. The measurements are performed at low $5<Q^2<100$ GeV$^2$ and high $150<Q^2<15000$ GeV$^2$ virtualities of the exchanged boson. The obtained cross sections are in good agreement with perturbative QCD calculations at NLO and are used to extract the strong coupling $\alpha_s(M_Z)$. The experimental precision is 0.6\% and 1.2\% for the high and low $Q^2$ regimes, respectively. The error on the obtained value of $\alpha_s(M_Z)$ is dominated by the theoretical uncertainties which are mostly due to missing higher orders. At low $Q^2$ the uncertainty due to terms beyond NLO amounts to 7.5\%. The determination of $\alpha_s$ at various scales shows the running of the strong coupling over a large range.