We analyze the oxygen abundances of a stellar sample representative of the two major Galactic populations: the thin and thick disks. The aim is to investigate the differences between members of the Galactic disks and to contribute to the understanding on the origin of oxygen chemical enrichment in the Galaxy. The analysis is based on the [O,{sc i}]=6300.30,AA~ oxygen line in HR spectra ($Rsim$52,500) obtained from the GES Survey. By comparing the observed spectra with a theoretical dataset, computed in LTE with the SPECTRUM synthesis and ATLAS12 codes, we derive the oxygen abundances of 516 FGK dwarfs for which we have previously measured carbon abundances. Based on kinematic, chemical and dynamical considerations we identify 20 thin and 365 thick disk members. We study potential trends of both subsamples in terms of their chemistry ([O/H], [O/Fe], [O/Mg], and [C/O] versus [Fe/H] and [Mg/H]), age, and position in the Galaxy. Main results are: (a) [O/H] and [O/Fe] ratios versus [Fe/H] show systematic differences between thin and thick disk stars with enhanced O abundance of thick disk stars with respect to thin disk members and a monotonic decrement of [O/Fe] with increasing metallicity, even at metal-rich regime; (b) a smooth correlation of [O/Mg] with age in both populations, suggesting that this abundance ratio can be a good proxy of stellar ages within the Milky Way; (c) thin disk members with [Fe/H]$simeq0$ display a [C/O] ratio smaller than the solar value, suggesting a possibly outward migration of the Sun from lower Galactocentric radii.
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