  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.
