We present ALMA band 8 observations of the [OIII] 88um line and the underlying thermal infrared continuum emission in the z=6.08 quasar CFHQS J2100-1715 and its dust-obscured starburst companion galaxy (projected distance: ~60 kpc). Each galaxy hosts dust-obscured star formation at rates > 100 M_sun/yr, but only the quasar shows evidence for an accreting 10^9 M_sun black hole. Therefore we can compare the properties of the interstellar medium in distinct galactic environments in two physically associated objects, ~1 Gyr after the Big Bang. Bright [OIII] 88um emission from ionized gas is detected in both systems; the positions and line-widths are consistent with earlier [CII] measurements, indicating that both lines trace the same gravitational potential on galactic scales. The [OIII] 88um/FIR luminosity ratios in both sources fall in the upper range observed in local luminous infrared galaxies of similar dust temperature, although the ratio of the quasar is smaller than in the companion. This suggests that gas ionization by the quasar (expected to lead to strong optical [OIII] 5008A emission) does not dominantly determine the quasars FIR [OIII] 88um luminosity. Both the inferred number of photons needed for the creation of O++ and the typical line ratios can be accounted for without invoking extreme (top-heavy) stellar initial mass functions in the starbursts of both sources.
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