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Magnetic fields are ubiquitous in active cool stars but they are in generalcomplex and weak. Current Zeeman Doppler imaging (ZDI) studies of cool starmagnetic fields chiefly employ circular polarization observations becauselinear polarization is difficult to detect and requires a more sophisticatedradiative transfer modeling to interpret. But it has been shown in previoustheoretical studies, and in the observational analyses of magnetic Ap stars,that including linear polarization in the magnetic inversion process makes itpossible to correctly recover many otherwise lost or misinterpreted magneticfeatures. We have obtained phase-resolved observations in all four Stokesparameters of the RS CVn star II Peg at two separate epochs. Here we presenttemperature and magnetic field maps reconstructed for this star using all fourStokes parameters. This is the very first such ZDI study of a cool active star.Our magnetic inversions reveal a highly structured magnetic field topology forboth epochs. The strength of some surface features is doubled or evenquadrupled when linear polarization is taken into account. The total magneticenergy of the reconstructed field map also becomes about 2.1-3.5 times higher.The overall complexity is also increased as the field energy is shifted towardshigher harmonic modes when four Stokes parameters are used. As a consequence,the potential field extrapolation of the four Stokes parameter ZDI resultsindicates that magnetic field becomes weaker at a distance of several stellarradii due to a decrease of the large-scale field component.

First Zeeman Doppler imaging of a cool star using all four Stokes parameters