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The aim is to investigate the differences between members of the Galactic disks and contribute to the understanding of the origin of oxygen chemical enrichment in the Galaxy. The analysis is based on the [O <jats:sc>i<\/jats:sc>]\u00a0=\u00a06300.30 \u00c5 oxygen line in high-resolution spectra (<jats:italic>R<\/jats:italic>\u00a0\u223c\u00a052,500) obtained from the Gaia-ESO public spectroscopic Survey (GES). By comparing the observed spectra with a theoretical data set 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 the 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. The main results are that (a) [O\/H] and [O\/Fe] ratios versus [Fe\/H] show systematic differences between thin and thick disk stars with an 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) there is 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; and (c) thin disk members with [Fe\/H]\u00a0\u2243\u00a00 display a [C\/O] ratio smaller than the solar value, suggesting a possibly outward migration of the Sun from lower Galactocentric radii.<\/jats:p>","DOI":"10.3847\/1538-3881\/abc69b","type":"journal-article","created":{"date-parts":[[2020,12,2]],"date-time":"2020-12-02T19:18:34Z","timestamp":1606936714000},"page":"9","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":16,"title":["The Gaia-ESO Survey: 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