Stars are fluid bodies in which pressure, buoyancy, rotation and magnetism provide the restoring forces for wave propagation. Constructive interference of these waves generates global modes of oscillation. With sufficiently precise photometry, stellar oscillations can be detected through the periodic variations they induce in a star's brightness. Understanding the connection between the internal structure of a star and the properties of its normal modes is the focus of stellar oscillation theory, which generally regards rotation and magnetism to be weak perturbations compared to pressure and buoyancy. The discovery several years ago that a sizable fraction of red giant stars exhibit abnormally low dipole mode amplitudes relative to the rest of the population (the "dipole dichotomy'' problem), suggesting the existence of a damping process localised to the deep interior, has presented a puzzle for theoreticians as it is not predicted in the standard framework. The restriction of this phenomenon to those stars with masses high enough to previously have hosted core dynamos points to the possible role of a hidden, relic magnetic field. It is the goal of the work presented herein to explore the nature of the interactions between waves in deep stellar interiors (in particular, gravity waves) and magnetic fields of dynamically significant strengths, that it might enable a better understanding of global mode properties and the ability to reconcile with observations.

The loss of spherical symmetry associated with the inclusion of magnetic fields renders it a mathematically complex problem to tackle, to which end we have invoked various complementary techniques: analytical treatment, numerical simulations and Hamiltonian ray tracing. We find that interactions between gravity waves and a magnetic field rely on a resonance criterion, and are always possible in some part of phase space, regardless of the field strength and the plasma $\beta$. Under these circumstances, waves may either become trapped within the field region and thereafter dissipated, or reflected in a near-specular fashion, the outcome depending on the relative orientations of the magnetic field lines, the wavevector, and the direction of stratification. Estimates of the damping rates associated with the trapping process, for realistic field configurations and field strengths of megagauss order, are consistent with those inferred from red giant observations, providing a promising resolution to the dipole dichotomy problem. The results in this thesis pave the way for an extension of the theory of stellar oscillations that incorporates the effects of dynamically significant magnetic fields.