Parametric Galois Extensions and Their Spectral Invariants

In this article we develop a dynamic formulation of Galois theory in which the classical symmetry group of a polynomial is replaced by a parametrized family of groups. The construction centers on a dependency space: a parameter set $\Gamma$ that determines, at each stage, which roots are admitted via a boundary condition in the base field. Each parameter $\gamma$ yields a restricted subfield and automorphism group, and the resulting family forms a projective system under canonical restriction maps. As the boundary varies, the system exhibits both smooth deformations and abrupt collapses of symmetry. We encode each group via its Cayley graph and use spectral invariants of the Laplacian to track structural changes: continuous eigenvalue variation signals stable phases, while jumps mark symmetry transitions. A quartic example demonstrates the framework explicitly.