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\section{Introduction}  Stellar magnetic fields at the stellar surface are routinely observed through their Zeeman spectral signature \citep{Landstreet_1992}. The magnetic flux can be inherited from the gas cloud from which the star is formed, or generated by a dynamo \citep{Brandenburg_2005}. A fraction of about 5-10\% of main sequence A stars is observed to have large scale, predominantly dipolar magnetic fields with amplitudes in the range 0.3-30kG \citep[Ap stars, see e.g.][]{Auri_re_2007}. These fields show little or no time evolution, which together with the existence of stable magnetic configurations \citep{Braithwaite_2006} supports the notion that they are not generated by a contemporary dynamo but rather inherited from the star formation phase (fossil fields).   Despite rapid progress in the discovery and characterization of surface magnetic fields, very little is known about internal stellar magnetic fields. This has prevented building a coherent picture of the evolution of stellar magnetism; for example it is not clear if the magnetic flux observed at the surface of main sequence stars can be conserved through the various phases of stellar evolution and be responsible for the existence of strongly magnetized compact remnants (magnetic WDs \citep{Wickramasinghe_2000} and magnetic NS ). \cite{Duncan_1992}).  After exhausting H in their cores, most main sequence stars evolve to become red giants.   During this phase the stellar structure is characterized by an expanding convective envelope and a contracting radiative core.