  Context. The influence of magnetic fields on stellar core collapse and
explosion is not well explored. It depends on the possibility to amplify the
pre-collapse fields. Without rotation this can happen by compression,
convection, the standing accretion shock instability, and the accumulation and
growth of Alfven waves in the accretion flows.
  Aims. We investigate such amplification mechanisms of the magnetic field
during the collapse and post-bounce evolution of the core of a non-rotating 15
solar mass star with varied initial field strengths, taking into account the
microphysical equation of state and neutrino physics crucial for supernova
cores.
  Methods. We perform simulations of ideal MHD with neutrino transport in
axisymmetry. The transport of electron neutrinos and antineutrinos is treated
with a new scheme that solves the energy-dependent set of radiation energy and
momentum equations in 2d by using an analytic closure relation.
  Results. The magnetic field undergoes amplification by turbulent flows. We
find indications for amplification by interacting waves in accretion streams.
The fields can reach up to equipartition with the velocity field. Very high
magnetic field strengths require very strong pre-collapse fields and are able
to shape the post-bounce flow, leading to a pattern dominated by low-order
multipoles. Such models are closest to a successful explosion.
  Conclusions. Magnetic fields can build up to interesting strengths even in
non-rotating collapsing stellar cores. Starting with fields in the pre-collapse
core as predicted by present stellar evolution models, typical neutron star
fields emerge, whereas stronger progenitor fields lead to fields of magnetar
strength. Only in the latter case the fields have dynamical effects on the
flows in the supernova core. However, in none of our 2D simulations, we find an
explosion until 500 ms post-bounce.
