Recent asteroseismic analyses have revealed the presence of strong (B $gtrsim 10^5$ G) magnetic fields in the cores of many red giant stars. Here, we examine the implications of these results for the evolution of stellar magnetic fields, and we make predictions for future observations. Those stars with suppressed dipole modes indicative of strong core fields should exhibit moderate but detectable quadrupole mode suppression. The long magnetic diffusion times within stellar cores ensure that dynamo-generated fields are confined to mass coordinates within the main sequence convective core, and the observed sharp increase in dipole mode suppression rates above $1.5 , M_odot$ may be explained by the larger convective core masses and faster rotation of these more massive stars. In clump stars, core fields of $sim10^5 , {rm G}$ can suppress dipole modes, whose visibility should be equal to or less than the visibility of suppressed modes in ascending red giants. High dipole mode suppression rates in low-mass ($M lesssim 2 , M_odot$) clump stars would indicate that magnetic fields generated during the main sequence can withstand subsequent convective phases and survive into the compact remnant phase. Finally, we discuss implications for observed magnetic fields in white dwarfs and neutron stars, as well as the effects of magnetic fields in various types of pulsating stars.
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