  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_{\nu} = 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_{\nu} = 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_{\nu} < 0.9$ eV (2
sigma C.L.), comparable to constraints obtained from the cosmic microwave
background data or other large scale structure probes.
