The intergalactic magnetic field within filaments should be less polluted by magnetized 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 adaptive mesh refinement simulations of magnetic fields in the cosmos and make predictions about the evolution and structure of magnetic fields in filaments. Comparing our results to observational evidence of magnetic fields in filaments suggests that amplification of seed fields by gravitational collapse is not sufficient to produce intergalactic medium fields. Finally, the implications for cosmic-ray transport are discussed.