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

Evolution of magnetic fields in galaxies in the frame of hierarchical structure formation cosmology: future tests with the SKA