Core magnetic field imprint in the non-radial oscillations of red giant
  stars
  Magnetic fields in red giant stars remain a poorly understood topic,
particularly in what concerns their intensity in regions far below the surface.
In this work, we propose that gravity-dominated mixed modes of high absolute
radial order and low angular degree can be used to probe the magnetic field in
their radiative cores. Using two poloidal, axisymmetric configurations for the
field in the core and the classical perturbative approach, we derive an
analytical expression for the magnetic frequency splitting of these oscillation
modes. Considering three distinct red giant models, with masses of
1.3\(M_\odot\), 1.6\(M_\odot\) and 2.0\(M_\odot\), we find that a field
strength of $10^5$ G is necessary in the core of these stars to induce a
frequency splitting of the order of a $\mu$Hz in dipole and quadrupole
oscillation modes. Moreover, taking into account observational limits, we
estimate that magnetic fields in the cores of red giants that do not present
observable magnetic splittings cannot exceed $10^4$ G. Given the general
absence of observable splittings in the oscillation spectra of these stars, and
assuming that present mode suppression mechanisms are not biased towards
certain azimuthal orders and retain all peaks in each multiplet, our results
lead us to conclude that internal fields with the considered configurations and
strengths above $10^4$ G are not prevalent in red giants.
