We report the detection of extended X-ray emission around two powerful high-z radio galaxies (HzRGs) at z~3.6 (4C03.24 & 4C19.71) and use these to investigate the origin of extended, Inverse Compton (IC) powered X-ray halos at high z. The halos have X-ray luminosities of Lx~3e44 erg/s and sizes of ~60kpc. Their morphologies are broadly similar to the ~60-kpc long radio lobes around these galaxies suggesting they are formed from IC scattering by relativistic electrons in the radio lobes, of either CMB or FIR photons from the dust-obscured starbursts in these galaxies. These observations double the number of z>3 HzRGs with X-ray detected IC halos. We compare the IC X-ray to radio luminosity ratios for these new detections to the two previously detected z~3.8 HzRGs. Given the similar redshifts, we would expect comparable X-ray IC luminosities if CMB mm photons are the seed field for the IC emission. Instead the two z~3.6 HzRGs, which are ~4x fainter in the FIR, also have ~4x fainter X-ray IC emission. Including a further six z>2 radio sources with IC X-ray halos from the literature, we suggest that in the more compact (lobe sizes <100-200kpc), majority of radio sources, the bulk of the IC emission may be driven by scattering of locally produced FIR photons from luminous, dust-obscured starbursts within these galaxies, rather than CMB photons. The resulting X-ray emission can ionise the gas on ~100-200-kpc scales around these systems and thus form their extended Ly-alpha emission line halos. The starburst and AGN activity in these galaxies are thus combining to produce an effective and wide-spread feedback process, acting on the long-term gas reservoir for the galaxy. If episodic radio activity and co-eval starbursts are common in massive, high-z galaxies, then this IC-feedback mechanism may affect the star-formation histories of massive galaxies. [Abridged]
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