  We use Monte Carlo simulations to identify the mechanism that allows for
phase transitions in dipolar spin ice to occur and survive for applied magnetic
field, H, much larger in strength than that of the spin-spin interactions. In
the most generic and highest symmetry case, the spins on one out of four
sublattices of the pyrochlore decouple from the total local
exchange+dipolar+applied field. In the special case where H is aligned
perfectly along the [110] crystallographic direction, spin chains perpendicular
to H show a transition to q=X long range order, which proceeds via a one to
three dimensional crossover. We propose that these transitions are relevant to
the origin of specific heat features observed in powder samples of the Dy2Ti2O7
spin ice material for H above 1 Tesla.
