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. Thompsons 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.
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