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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 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 $B \sim 10^4 {\rm G}$).  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 (***REF***). \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.).