The magnetic field topology plays an important role in the understanding of stellar magnetic activity . While it is widely accepted that the dynamo action present in low-mass partially convective stars ( e.g. , the Sun ) results in predominantly toroidal magnetic flux , the field topology in fully convective stars ( masses below \sim 0.35 M _ { \odot } ) is still under debate . 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 \pm 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 .