  The protons in water ice are subject to so called \emph{ice rules} resulting
in an extensive ground state degeneracy. We study how an external electric
field reduces this ground state degeneracy in hexagonal water ice I$_h$ within
a minimal model. We observe polarization plateaus when the field is aligned
along the $[001]$ and $[010]$ directions. In each case, one plateau occurs at
intermediate polarization with reduced but still extensive degeneracy. The
remaining ground states can be mapped to dimer models on the honeycomb and the
square lattice, respectively. Upon tilting the external field, we observe an
order-disorder transition of Kasteleyn type into a plateau at saturated
polarization and vanishing entropy. This transition is investigated
analytically using the Kasteleyn matrix and numerically using a modified
directed-loop Monte Carlo simulation. The protons in both cases exhibit
algebraically decaying correlations. Moreover, the features of the static
structure factor are discussed.
