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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. 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}. Core 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. These strong fields may reside within the inner core with little external manifestation apart from the suppressed visibility of the dipole modes.  In the early red giant KIC 8561221 (\cite{Garcia_2014}), we have calculated a radial field strength $B_r \approx 1.5 \times 10^7 \, {\rm G}$ at the H-burning shell. This star demonstrates that very strong magnetic fields ($B \gg 10^6 \, {\rm G}$) can exist within the radiative cores of early red giant stars. Since these fields are likely inherited from previous stages of stellar evolution, slightly weaker ($B \gg 10^5 \,{\rm G})$ fields likely exist within the cores of some main sequence stars.  %(sub-equipartition magnetic fields are easily above $B \sim 10^4 {\rm G}$).