  We investigate with computer simulations the effect of applying an electric
field on the water-to-ice transition. We use a combination of state-of-the-art
simulation techniques to obtain phase boundaries and crystal growth rates
(direct coexistence), nucleation rates (seeding) and interfacial free energies
(seeding and mold integration). First, we consider ice Ih, the most stable
polymorph in the absence of a field. Its normal melting temperature, speed of
crystal growth and nucleation rate (for a given supercooling) diminish as the
intensity of the field goes up. Then, we study polarised cubic ice, or ice Icf,
the most stable solid phase under a strong electric field. Its normal melting
point goes up with the field and, for a given supercooling, under the studied
field (0.3 V/nm) ice Icf nucleates and grows at a similar rate as Ih with no
field. The net effect of the field would be then that ice nucleates at warmer
temperatures, but in the form of ice Icf. The main conclusion of this work is
that reasonable electric fields (not strong enough to break water molecules
apart) are not relevant in the context of homogeneous ice nucleation at 1 bar.
