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We have shown that the magnetic greenhouse effect is an effective wave trapping mechanism, thanks to the symmetry breaking enforced by any plausible geometry of a (strong enough) magnetic field within the core. For perfect wave trapping, purely dipole modes only exist in the envelope, with part of their energy leaking into the core as running magneto-gravity waves. If some wave energy does escape the core, it may leave a signature in the form of mixed magneto-gravity acoustic modes which could be used to constrain the internal magnetic field geometry.  %Hence, mixed modes in the usual sense do not exist in stars with suppressed dipole modes.  In principle, it is possible that another symmetry breaking mechanism could suppress dipole mode amplitudes. The only other plausible candidate is rapid core rotation, but we argue this is an unlikely explanation for most suppressed pulsators (see details in the supplementary material). rotation.  In order for rotation to strongly modify the incoming waves such that they will be trapped in the core, the core must rotate at a frequency comparable to $\nu_{\rm max}$, roughly two orders of magnitude faster than the values commonly observed in the cores of these stars \citep{Beck_2011,Mosser_2012}. The suppressed pulsator KIC 8561221 (\cite{Garcia_2014}) does not exhibit rapid envelope rotation and disfavors the rotation scenario.Moreover the magnetic greenhouse effect makes a clear prediction: that for stars with frequency of maximum power similar to the critical magneto gravity frequency $\nu_{\rm c}$, dipole modes with $\nu >\nu_{\rm c}$ will be unaffected, while those with $\nu <\nu_{\rm c}$ should show suppression. KIC 8561221 displays this exact behavior, supporting the magnetic greenhouse mechanism.  %but we argue this is an unlikely explanation for most suppressed pulsators (see details in the supplementary material). Moreover the magnetic greenhouse effect makes a clear prediction: that for stars with frequency of maximum power similar to the critical magneto gravity frequency $\nu_{\rm c}$, dipole modes with $\nu >\nu_{\rm c}$ will be unaffected, while those with $\nu <\nu_{\rm c}$ should show suppression. KIC 8561221 displays this exact behavior, supporting the magnetic greenhouse mechanism.  For the magnetic greenhouse effect to operate, stars must have magnetic fields with a radial component $B\gtrsim 10^4 {\rm G}$ (see Figure \ref{fig:Bc}) around the location of the H-burning shell. We note that magnetic fields of similar amplitude have been discussed in order to explain the suppression of thermohaline mixing in a small fraction of red giant stars, as inferred from the observations of their surface abundances \cite{Charbonnel_2007}. We show (see supplementary material) that magnetic Magnetic  fields with these characteristics could be present if the star retained a fossil field with surface amplitude $\sim 1 {\rm kG}$ on the main sequence, or if a convective core dynamo was at work during the main sequence (sub-equipartition magnetic fields are easily above $B \sim 10^4 {\rm G}$). %We show (see supplementary material) that  The fraction of stars showing suppressed dipole modes in the sample of \citet{Mosser_2011} is about $22\%$, suggesting that red giants with magnetized cores are not just the descendants of magnetic Ap stars, which comprise less than $\sim \! 10 \%$ of A type stars \citep{Wolff_1968}. A detailed analysis of a large population of red giants with suppressed dipole modes will put strong constraints on the amplitude and evolution of internal magnetic fields in stars of different masses (Stello et al. in prep.).  The asteroseismic technique described in this paper can also be applied to red clump stars burning helium in their cores. Observations of dipole mode suppression (or lack thereof) in clump stars will also put important constraints on the internal magnetic fields of the immediate progenitors of white dwarfs (AuthorX et al. In prep.).