  We present non-radiative, cosmological zoom-simulations of galaxy cluster
formation with magnetic fields and (anisotropic) thermal conduction of one very
massive galaxy cluster with a mass at redshift zero that corresponds to
$M_\mathrm{vir} \sim 2 \times 10^{15} M_{\odot}$. We run the cluster on three
resolution levels (1X, 10X, 25X), starting with an effective mass resolution of
$2 \times 10^8M_{\odot}$, subsequently increasing the particle number to reach
$4 \times 10^6M_{\odot}$. The maximum spatial resolution obtained in the
simulations is limited by the gravitational softening reaching $\epsilon=1.0$
kpc at the highest resolution level, allowing to resolve the hierarchical
assembly of the structures in very fine detail. All simulations presented, have
been carried out with the SPMHD-code Gadget-3 with a heavily updated SPMHD
prescription. The primary focus is to investigate magnetic field amplification
in the Intracluster Medium (ICM). We show that the main amplification mechanism
is the small scale-turbulent-dynamo in the limit of reconnection diffusion. In
our two highest resolution models we start to resolve the magnetic field
amplification driven by this process and we explicitly quantify this with the
magnetic power-spectra and the magnetic tension that limits the bending of the
magnetic field lines consistent with dynamo theory. Furthermore, we investigate
the $\nabla \cdot \mathbf{B}=0$ constraint within our simulations and show that
we achieve comparable results to state-of-the-art AMR or moving-mesh
techniques, used in codes such as Enzo and Arepo. Our results show for the
first time in a fully cosmological simulation of a galaxy cluster that dynamo
action can be resolved in the framework of a modern Lagrangian
magnetohydrodynamic (MHD) method, a study that is currently missing in the
literature.
