  Abundance ratios in extremely metal-poor (EMP) stars are a good indication of
the chemical composition of the gas in the earliest phases of the Galaxy
evolution. It had been found from an LTE analysis that at low metallicity, and
in contrast with most of the other elements, the scatter of [Na/Fe] versus
[Fe/H] was surprisingly large and that, in giants, [Na/Fe] decreased with
metallicity.
  Since it is well known that the formation of sodium lines is very sensitive
to non-LTE effects, to firmly establish the behaviour of the sodium abundance
in the early Galaxy, we have used high quality observations of a sample of EMP
stars obtained with UVES at the VLT, and we have taken into account the non-LTE
line formation of sodium.
  The profiles of the two resonant sodium D lines (only these sodium lines are
detectable in the spectra of EMP stars) have been computed in a sample of 54
EMP giants and turn-off stars (33 of them with [Fe/H]< -3.0) with a modified
version of the code MULTI, and compared to the observed spectra.
  With these new determinations in the range -4 <[Fe/H]< -2.5, both [Na/Fe] and
[Na/Mg] are almost constant with a low scatter. In the turn-off stars and
"unmixed" giants (located in the low RGB): [Na/Fe] = -0.21 +/- 0.13 or [Na/Mg]
= -0.45 +/- 0.16. These values are in good agreement with the recent
determinations of [Na/Fe] and [Na/Mg] in nearby metal-poor stars. Moreover we
confirm that all the sodium-rich stars are "mixed" stars (i.e., giants located
after the bump, which have undergone an extra mixing). None of the turn-off
stars is sodium-rich. As a consequence it is probable that the sodium
enhancement observed in some mixed giants is the result of a deep mixing.
