The intergalactic magnetic field within filaments should be less polluted by magnetised outflows from active galaxies than magnetic fields in clusters. Therefore, filaments may be a better laboratory to study magnetic field amplification by structure formation than galaxy clusters which typically host many more active galaxies. We present highly resolved cosmological AMR simulations of magnetic fields in the cosmos and make predictions about the evolution and structure of magnetic fields in

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... ments. Comparing our results to observational evidence for magnetic fields in filaments suggests that amplification of seed fields by gravitational collapse is not sufficient to produce IGM fields. Finally, implications for cosmic ray transport are discussed.