  While emission and absorption lines of the 21 cm spin-flip transition of
neutral Hydrogen are intrinsically unpolarized, a magnetic field creates left-
and right-handed polarized components through the Zeeman effect. Here we
consider the resulting polarization of the redshifted 21 cm background from the
intergalactic medium before reionization. The polarization is detectable in
regions with a strong gradient in the mean brightness temperature. In
principle, this can open a new window on the evolution of intergalactic
magnetic fields. One possible approach is an extended integration of an
individual target during this era, such as the Mpc-scale HII regions inferred
to surround quasars at z ~ 6.5. The differential intensity between the two
polarization states can be used as a probe of the magnetic field at the edge of
the HII region. We estimate that the SKA could (ignoring systematics) detect B
\~ 200 (10) \muG coherent over several kiloparsecs with an observational
bandwidth of 100 (2) kHz. Beyond individual sources, the statistical properties
of wide-field 21 cm polarization maps, such as the angular power spectrum, can
be used to constrain the large-scale magnetic field. In this case, the SKA can
detect $B \sim 100 \mu$G fields coherent over many megaparsecs. The magnetic
field can be measured in any epoch over which the 21 cm background changes
rapidly (for example because the ionized fraction or spin temperature change).
Although the resulting constraints with SKA are relatively weak compared to
theoretical expectations, they nevertheless offer a unique direct probe of
magnetic fields in the high-redshift universe.
