(abridged) We investigate the possibility to recognize the magnetic field structures in nearby galaxies and to test the cosmological evolution of their large- and small-scale magnetic fields with the SKA and its precursors. We estimate the required density of the background polarized sources detected with the SKA for reliable reconstruction and reconstruction of magnetic field structures in nearby spiral galaxies. The dynamo theory is applied to distant galaxies to explore the evolution of magnetic fields in distant galaxies in the context of a hierarchical dark matter cosmology. Under favorite conditions, a emph{recognition} of large-scale magnetic structures in local star-forming disk galaxies (at a distance $la 100$ Mpc) is possible from $ga 10$ RMs towards background polarized sources. Galaxies with strong turbulence or small inclination need more polarized sources for a statistically reliable recognition. A reliable emph{reconstruction} of the field structure without precognition needs at least 20 RM values on a cut along the projected minor axis which translates to $approx1200$ sources towards the galaxy. We demonstrate that early regular fields are already in place at $z sim 4$ (approximately 1.5 Gyr after the disk formation) in massive gas-rich galaxies ($>10^9$ M$_{sun}$) which then evolve to Milky-Way type galaxies. Major and minor mergers influence the star formation rate and geometry of the disk which has an effect of shifting the generation of regular fields in disks to later epochs. Predictions of the evolutionary model of regular fields, simulations of the evolution of turbulent and large-scale regular fields, total and polarized radio emission of disk galaxies, as well as future observational tests with the SKA are discussed.
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