  We present sensitive, high-resolution, X-ray imaging from Chandra of the
high-redshift radio galaxy 4C 41.17 (z=3.8). Our 150-ks Chandra exposure
detects strong X-ray emission from a point source coincident with the nucleus
of the radio galaxy. In addition we identify extended X-ray emission with a
luminosity ~1e45 erg/s covering a 100kpc (15'') diameter region around the
radio galaxy. The extended X-ray emission follows the general distribution of
radio emission in the radio lobes of this source, and the distribution of a
giant Lyman-alpha emission line halo, while the spectrum of the X-ray emission
is non-thermal and has a power law index consistent with that of the radio
synchrotron. We conclude that the X-ray emission is most likely Inverse-Compton
scattering of far-infrared photons from a relativistic electron population
probably associated with past and current activity from the central object.
Assuming an equipartition magnetic field the CMB energy density at z=3.8 can
only account for at most 40% of the Inverse-Compton emission. Published
submillimeter maps of 4C 41.17 have detected an apparently extended and
extremely luminous far-infrared emission around the radio galaxy. We
demonstrate that this photon component and its spatial distribution, in
combination with the CMB can reproduce the observed X-ray luminosity. We
propose that photo-ionization by these Inverse-Compton X-ray photons plays a
significant role in this system, and provides a new physical feedback mechanism
to preferentially affect the gas within the most massive halos at high
redshift. This is the highest redshift example of extended X-ray emission
around a radio galaxy currently known. (Abridged)
