We present a suite of full hydrodynamical cosmological simulations that quantitatively address the impact of neutrinos on the (mildly non-linear) spatial distribution of matter and in particular on the neutral hydrogen distribution in the Intergalactic Medium (IGM), which is responsible for the intervening Lyman-alpha absorption in quasar spectra. The free-streaming of neutrinos results in a (non-linear) scale-dependent suppression of power spectrum of the total matter distribution at scales probed by Lyman-alpha forest data which is larger than the linear theory prediction by about 25% and strongly redshift dependent. By extracting a set of realistic mock quasar spectra, we quantify the effect of neutrinos on the flux probability distribution function and flux power spectrum. The differences in the matter power spectra translate into a ~2.5% (5%) difference in the flux power spectrum for neutrino masses with Sigma m_{ u} = 0.3 eV (0.6 eV). This rather small effect is difficult to detect from present Lyman-alpha forest data and nearly perfectly degenerate with the overall amplitude of the matter power spectrum as characterised by sigma_8. If the results of the numerical simulations are normalized to have the same sigma_8 in the initial conditions, then neutrinos produce a smaller suppression in the flux power of about 3% (5%) for Sigma m_{ u} = 0.6$ eV (1.2 eV) when compared to a simulation without neutrinos. We present constraints on neutrino masses using the Sloan Digital Sky Survey flux power spectrum alone and find an upper limit of Sigma m_{ u} < 0.9$ eV (2 sigma C.L.), comparable to constraints obtained from the cosmic microwave background data or other large scale structure probes.
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