  Results from simulations of hierarchical structure formation cosmology
provide a tool to develop an evolutionary model of regular magnetic fields
coupled to galaxy formation and evolution. We use the dynamo theory to derive
the timescales of amplification and ordering of magnetic fields in disk and
puffy galaxies. Galaxies similar to the Milky Way formed their disks at
$z\approx10$ and regular fields of $\mu$G strength and a few kpc coherence
length were generated within 2 Gyr (at $z\approx3$), but field ordering up to
the coherence scale of the galaxy size took another 6 Gyr (at $z\approx0.5$).
Giant galaxies formed their disk already at $z\approx10$, allowing more
efficient dynamo generation of strong regular fields (with kpc coherence
length) already at $z\approx4$. Dwarf galaxies should have hosted fully
coherent fields at $z\approx1$. This evolutionary scenario and number of
predictions of the model can be tested by measurements of polarized synchrotron
emission and Faraday rotation with the planned Square Kilometre Array. This
model is used to simulate the evolution of regular fields in disk galaxies and
the polarized radio sky as part of the Square Kilometer Array Design Studies
(SKADS).
