  Fast radio bursts (FRBs) are millisecond-duration, extragalactic radio
flashes of unknown physical origin. FRB 121102, the only known repeating FRB
source, has been localized to a star-forming region in a dwarf galaxy at
redshift z = 0.193, and is spatially coincident with a compact, persistent
radio source. The origin of the bursts, the nature of the persistent source,
and the properties of the local environment are still debated. Here we present
bursts that show ~100% linearly polarized emission at a very high and variable
Faraday rotation measure in the source frame: RM_src = +1.46 x 10^5 rad m^-2
and +1.33 x 10^5 rad m^-2 at epochs separated by 7 months, in addition to
narrow (< 30 mus) temporal structure. The large and variable rotation measure
demonstrates that FRB 121102 is in an extreme and dynamic magneto-ionic
environment, while the short burst durations argue for a neutron star origin.
Such large rotation measures have, until now, only been observed in the
vicinities of massive black holes (M_BH > 10^4 MSun). Indeed, the properties of
the persistent radio source are compatible with those of a low-luminosity,
accreting massive black hole. The bursts may thus come from a neutron star in
such an environment. However, the observed properties may also be explainable
in other models, such as a highly magnetized wind nebula or supernova remnant
surrounding a young neutron star.
