  Stars more massive than $\sim 1.3$ M$_\odot$ are known to develop a
convective core during the main-sequence: the dynamo process triggered by this
convection could be the origin of a strong magnetic field inside the core of
the star, trapped when it becomes stably stratified and for the rest of its
evolution. The presence of highly magnetized white dwarfs strengthens the
hypothesis of buried fossil magnetic fields inside the core of evolved low-mass
stars. If such a fossil field exists, it should affect the mixed modes of red
giants as they are sensitive to processes affecting the deepest layers of these
stars. The impact of a magnetic field on dipolar oscillations modes was one of
Pr. Michael J. Thompson's research topics during the 90s when preparing the
helioseismic SoHO space mission. As the detection of gravity modes in the Sun
is still controversial, the investigation of the solar oscillation modes did
not provide any hint of the existence of a magnetic field in the solar
radiative core. Today we have access to the core of evolved stars thanks to the
asteroseismic observation of mixed modes from CoRoT, Kepler, K2 and TESS
missions. The idea of applying and generalizing the work done for the Sun came
from discussions with Pr. Michael Thompson in early 2018 before we loss him.
Following the path we drew together, we theoretically investigate the effect of
a stable axisymmetric mixed poloidal and toroidal magnetic field, aligned with
the rotation axis of the star, on the mixed modes frequencies of a typical
evolved low-mass star. This enables us to estimate the magnetic perturbations
to the eigenfrequencies of mixed dipolar modes, depending on the magnetic field
strength and the evolutionary state of the star. We conclude that strong
magnetic fields of $\sim$ 1MG should perturbe the mixed-mode frequency pattern
enough for its effects to be detectable inside current asteroseismic data.
