  We present a study of the vertical magnetic field of the Milky Way towards
the Galactic poles, determined from observations of Faraday rotation toward
more than 1000 polarized extragalactic radio sources at Galactic latitudes |b|
> 77 degs, using the Westerbork Radio Synthesis Telescope and the Australia
Telescope Compact Array. We find median rotation measures (RMs) of 0.0 +/- 0.5
rad/m^2 and +6.3 +/- 0.7 rad/m^2 toward the north and south Galactic poles,
respectively, demonstrating that there is no coherent vertical magnetic field
in the Milky Way at the Sun's position. If this is a global property of the
Milky Way's magnetism, then the lack of symmetry across the disk rules out pure
dipole or quadrupole geometries for the Galactic magnetic field. The angular
fluctuations in RM seen in our data show no preferred scale within the range ~
0.1 to 25 degs. The observed standard deviation in RM of ~ 9 rad/m^2 then
implies an upper limit of ~1microGauss on the strength of the random magnetic
field in the warm ionized medium at high Galactic latitudes.
