Measurements are presented of the t-channel single-top-quark production cross section in proton-proton collisions at root s = 8 TeV. The results are based on a data sample corresponding to an integrated luminosity of 19.7 fb(-1) recorded with the CMS detector at the LHC. The cross section is measured inclusively, as well as separately for top (t) and antitop (t) over bar, in final states with a muon or an electron. The measured inclusive t-channel cross section is sigma(t-ch.) = 83.6 +/- 2.3 (stat.) +/- 7.4 (syst.) pb. The single t and (t) cross sections are measured to be sigma(t-ch.)(t) = 53.8 +/- 1.5 (stat.) +/- 4.4 (syst.) pb and sigma(t-ch.)(t) over bar = 27.6 +/- 1.3 (stat.) +/- 3.7 (syst.) pb, respectively. The measured ratio of cross sections is Rt-ch. = sigma(t-ch.)(t)/sigma(t-ch.)(t) over bar = 1.95 +/- 0.10 (stat.) +/- 0.19 (syst.), in agreement with the standard model prediction. The modulus of the Cabibbo-Kobayashi-Maskawa matrix element V-tb is extracted and, in combination with a previous CMS result at root s = 7 TeV, a value vertical bar V-tb vertical bar = 0.998 +/- 0.038 (exp.) +/- 0.016 (theo.) is obtained.