  We investigate the cosmological evolution of large- and small-scale magnetic
fields in galaxies at high redshifts. 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.
Turbulence in protogalactic halos generated by thermal virialization can drive
an efficient turbulent dynamo. The mean-field dynamo theory is used to derive
the timescales of amplification and ordering of regular magnetic fields in disk
and dwarf galaxies. For future observations with the SKA, we predict an
anticorrelation at fixed redshift between galaxy size and the ratio between
ordering scale and galaxy size. Undisturbed dwarf galaxies should host fully
coherent fields at z<1, spiral galaxies at z<0.5.
