  (abridged) We suggest here that the large scale fields $\sim \mu$G, observed
in galaxies at both high and low redshifts by Faraday Rotation Measurements
(FRMs), have their origin in the electromagnetic fluctuations that naturally
occurred in the dense hot plasma that existed just after the QHPT. We evolve
the predicted fields to the present time. The size of the region containing a
coherent magnetic field increased due to the fusion (polymerization) of smaller
regions. Magnetic fields (MFs) $\sim 10 \mu G$ over a comoving $\sim 1$ pc
region are predicted at redshift z $\sim 10$. These fields are orders of
magnitude greater than those predicted in previous scenarios for creating
primordial magnetic fields. Line-of-sight average magnetic fields (MFs) $\sim$
$10^{-2}$ $\mu$G, valid for FRMs, are obtained over a 1 Mpc comoving region at
the redshift z $\sim$ 10.
  In the collapse to a galaxy (comoving size $\sim$ 30 kpc) at z $\sim$ 10, the
fields are amplified to $\sim 10 \mu$G. This indicates that the MFs created
immediately after the QHPT, predicted by the Fluctuation-Dissipation Theorem,
could be the origin of the $\sim \mu G$ fields observed by FRMs in galaxies at
both high and low redshifts.
