  Lines of diatomic molecules are more temperature and pressure sensitive than
atomic lines, which makes them ideal tools for studying cool stellar
atmospheres an internal structure of sunspots and starspots. The FeH F^4
Delta-X^4 Delta system represents such an example that exhibits in addition a
large magnetic field sensitivity. The current theoretical descriptions of these
transitions including the molecular constants involved are only based on
intensity measurements because polarimetric observations have not been
available so far, which limits their diagnostic value. We present for the first
time spectropolarimetric observations of the FeH F^4 Delta-X^4 Delta system
measured in sunspots to investigate their diagnostic capabilities for probing
solar and stellar magnetic fields. We investigate whether the current
theoretical model of FeH can reproduce the observed Stokes profiles including
their magnetic properties. The polarimetric observations are compared with
synthetic Stokes profiles modeled with radiative transfer calculations. This
allows us to infer the temperature and the magnetic field strength of the
observed sunspots. We find that the current theory successfully reproduces the
magnetic properties of a large number of lines in the FeH F^4 Delta-X^4 Delta
system. In a few cases the observations indicate a larger Zeeman splitting than
predicted by the theory. There, our observations have provided additional
constraints, which allowed us to determine empirical molecular constants. The
FeH F^4 Delta-X^4 Delta system is found to be a very sensitive magnetic
diagnostic tool. Polarimetric data of these lines provide us with more direct
information to study the coolest parts of astrophysical objects.
