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.
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