Measurements of the jet and vector boson in association with jets production, extraction of $\alpha_s$ and PDF constraints at CMS

We present CMS results about measurements of cross sections for jet production, which allow basic tests of perturbative QCD predictions. Results include recent jet, dijet, and multijet differential cross sections, and hadronic event shape measurements in pp collisions at a centre-of-mass energy of 7 and 8 TeV, using data collected in the CMS experiment. Total and differential cross section measurements of vector bosons production in association with jets and heavy flavour quarks are also presented. The experimental results are compared to leading and next-to-leading order calculations, both at parton level and in event generators where matrix element calculations are interfaced with parton showers. Recent CMS results related to the extraction of the strong coupling constant and extraction of alpha_S and PDF are discussed.