  The origin and the magnitude of the inter-galactic magnetic field is of
primordial importance in the global picture of magnetic field evolution, as it
is considered to be the missing link between galactic magnetic fields and
cluster magnetic fields on much larger scales. We are testing whether dwarf
galaxies are good candidates to explain the enrichment of the IGM: after their
discs form and trigger galactic dynamos, supernova feedback will launch strong
winds, expelling magnetic field lines in the IGM. We have performed MHD
simulations of an isolated dwarf galaxy, forming self-consistently inside a
cooling halo. Using the RAMSES code, we have for the first time simulated the
formation of a magnetised supernova-driven galactic outflow. This simulation is
an important step towards a more realistic modelling using fully cosmological
simulations. Our simulations reproduce well the observed properties of magnetic
fields in spiral galaxies. The formation and the evolution of our simulated
disc lead to a strong magnetic field amplification: the magnetic field in the
final wind bubble is one order of magnitude larger than the initial value. The
magnetic field in the disc, essentially toroidal, is growing linearly with time
as a consequence of differential rotation. We discuss the consequence of this
simple mechanism on the cosmic evolution of the magnetic field: we propose a
new scenario for the evolution of the magnetic field, with dwarf galaxies
playing a key role in amplifying and ejecting magnetic energy in the IGM,
resulting in what we call a "Cosmic Dynamo" that could contribute to the rather
high field strengths observed in galaxies and clusters today.
