A Galois-Siegel Grand Theory for Ramanujan's Chapter-20 Prime Dissections: Phase Descent, Fixed Fields, and the Auxiliary Equations

We give a uniform structural interpretation of Ramanujan's Chapter-20 prime-dissection remark. For every prime p ≥ 5, the normalized p-th radicands arising from the Euler function f and Ramanujan's theta functions phi and psi generate one common cyclic modular-function extension with Galois group (Z/pZ)^×/{±1}. We prove that the p-primary phase obstruction is supported on the faithful cyclotomic component, while subgroup descent annihilates this component and produces the corresponding fixed fields. Explicit primitive relative-trace and Hilbert-90 packet coordinates yield genuine prime-degree auxiliary equations. This gives a uniform mechanism for Ramanujan's quadratic, cubic, and quintic prime lists, with historical representative realizations at p = 5, 13, and 11. External specialist correctness, priority, and historical-interpretation review have not yet occurred.