We study the Oxygen and Nitrogen abundances in the interstellar medium of high-redshift galaxies. We use high resolution and high signal-to-noise ratio spectra of Damped Lyman-alpha (DLA) systems detected along the line-of-sight to quasars to derive robust abundance measurements from unsaturated metal absorption lines. We present results for a sample of 16 high-redshift DLAs and strong sub-DLAs (log N(HI)>19.5, 2.4<zabs<3.6) including 13 new measurements. We find that the Oxygen to Iron abundance ratio is pretty much constant with [O/Fe]=+0.32+-0.10 for -2.5<[O/H]<-1.0 with a small scatter around this value. The Oxygen abundance follows quite well the Silicon abundance within 0.2dex although the Silicon abundance could be slightly smaller for [O/H]<-2. The distribution of the [N/O] abundance ratio, measured from components that are detected in both species, is somehow double peaked: five systems have [N/O]>-1 and nine systems have [N/O]<-1.15. In the diagram [N/O] versus [O/H], a loose plateau is possibly present at [N/O]=-0.9 that is below the so-called primary plateau as seen in local metal-poor dwarf galaxies ([N/O] in the range -0.57 to -0.74). No system is seen above this primary plateau whereas the majority of the systems lie well below with a large scatter. All this suggests a picture in which DLAs undergo successive star-bursts. During such an episode, the [N/O] ratio decreases sharply because of the rapid release of Oxygen by massive stars whereas inbetween two bursts, Nitrogen is released by low and intermediate-mass stars with a delay and the [N/O] ratio increases.
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