  We discuss the evolution of the magnetic field of an accreting white dwarf.
We first show that the timescale for ohmic decay in the liquid interior is 8 to
12 billion years for a dipole field, and 4 to 6 billion years for a quadrupole
field. We then compare the timescales for ohmic diffusion and accretion at
different depths in the star, and for a simplified field structure and
spherical accretion, calculate the time-dependent evolution of the global
magnetic field at different accretion rates. In this paper, we neglect mass
loss by classical nova explosions and assume the white dwarf mass increases
with time. In this case, the field structure in the outer layers of the white
dwarf is significantly modified for accretion rates above the critical rate
(1-5) x 10^(-10) solar masses per year. We consider the implications of our
results for observed systems. We propose that accretion-induced magnetic field
changes are the missing evolutionary link between AM Her systems and
intermediate polars. The shorter ohmic decay time for accreting white dwarfs
provides a partial explanation of the lack of accreting systems with 10^9 G
fields. In rapidly accreting systems such as supersoft X-ray sources,
amplification of internal fields by compression may be important for Type Ia
supernova ignition and explosion. Finally, spreading matter in the polar cap
may induce complexity in the surface magnetic field, and explain why the more
strongly accreting pole in AM Her systems has a weaker field. We conclude with
speculations about the field evolution when classical nova explosions cause the
white dwarf mass to decrease with time.
