  We present a second-order Green's-function theory of the one- and
two-dimensional S=1/2 ferromagnet in a magnetic field based on a decoupling of
three-spin operator products, where vertex parameters are introduced and
determined by exact relations. The transverse and longitudinal spin correlation
functions and thermodynamic properties (magnetization, isothermal magnetic
susceptibility, specific heat) are calculated self-consistently at arbitrary
temperatures and fields. In addition, exact diagonalizations on finite lattices
and, in the one-dimensional case, exact calculations by the Bethe-ansatz method
for the quantum transfer matrix are performed. A good agreement of the
Green's-function theory with the exact data, with recent quantum Monte Carlo
results, and with the spin polarization of a $\nu=1$ quantum Hall ferromagnet
is obtained. The field dependences of the position and height of the maximum in
the temperature dependence of the susceptibility are found to fit well to power
laws, which are critically analyzed in relation to the recently discussed
behavior in Landau's theory. As revealed by the spin correlation functions and
the specific heat at low fields, our theory provides an improved description of
magnetic short-range order as compared with the random phase approximation. In
one dimension and at very low fields, two maxima in the temperature dependence
of the specific heat are found. The Bethe-ansatz data for the field dependences
of the position and height of the low-temperature maximum are described by
power laws. At higher fields in one and two dimensions, the temperature of the
specific heat maximum linearly increases with the field.
