The discovery of the moderate differential rotation between the core and the envelope of evolved solar-like stars could be the signature of a strong magnetic field trapped inside the radiative interior. The population of intermediate-mass red giants presenting a surprisingly low-amplitude of their mixed modes could also arise from the effect of an internal magnetic field. Indeed, stars more massive than about 1.1Ms are known to develop a convective core during their main sequence, which could relax into a strong fossil magnetic field trapped inside the core of the star for the rest of its evolution. The observations of mixed modes can constitute an excellent probe of the deepest layers of evolved solar-like stars. The magnetic perturbation on mixed modes may thus be visible in asteroseismic data. To unravel which constraints can be obtained from observations, we theoretically investigate the effects of a plausible mixed axisymmetric magnetic field with various amplitudes on the mixed-mode frequencies of evolved solar-like stars. The first-order frequency perturbations are computed for dipolar and quadrupolar mixed modes. These computations are carried out for a range of stellar ages, masses, and metallicities. We show that typical fossil-field strengths of 0.1-1 MG, consistent with the presence of a dynamo in the convective core during the main sequence, provoke significant asymmetries on mixed-mode frequency multiplets during the red-giant branch. We show that these signatures may be detectable in asteroseismic data for field amplitudes small enough for the amplitude of the modes not to be affected by the conversion of gravity into Alfven waves inside the magnetised interior. Finally, we infer an upper limit for the strength of the field, and the associated lower limit for the timescale of its action, to redistribute angular momentum in stellar interiors.
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