  (ABRIDGED) We report here our mapping of the magnetic field topology of the
M4 dwarf G 164-31 (or Gl 490B), which is expected to be fully convective, based
on time series data collected from 20 hours of observations spread over 3
successive nights with the ESPaDOnS spectropolarimeter. Our tomographic imaging
technique applied to time series of rotationally modulated circularly polarized
profiles reveals an axisymmetric large-scale poloidal magnetic field on the M4
dwarf. We then apply a synthetic spectrum fitting technique for measuring the
average magnetic flux on the star. The flux measured in G 164-31 is Bf =
3.2+-0.4 kG, which is significantly greater than the average value of 0.68 kG
determined from the imaging technique. The difference indicates that a
significant fraction of the stellar magnetic energy is stored in small-scale
structures at the surface of G 164-31. Our H_alpha emission light curve shows
evidence for rotational modulation suggesting the presence of localized
structure in the chromosphere of this M dwarf. The radius of the M4 dwarf
derived from the rotational period and the projected equatorial velocity is at
least 30% larger than that predicted from theoretical models. We argue that
this discrepancy is likely primarily due to the young nature of G 164-31 rather
than primarily due to magnetic field effects, indicating that age is an
important factor which should be considered in the interpretation of this
observational result. We also report here our polarimetric observations of five
other M dwarfs with spectral types from M0 to M4.5, three of them showing
strong Zeeman signatures.
