Algebraic Geometry of Norm Tori over Finite Fields and the Unified Solution of Higher-Degree Congruence Equations

This paper aims to establish an algebro-geometric framework for solving the n-th degree congruence equation xⁿ p over the finite field Fₚ, transforming the traditional arithmetic determination of residue classes into a problem concerning the intersection of algebraic varieties with the base field. First, on the algebraic torus R₅_₏ℂ/Fₚ (Gₘ) of the quadratic extension, using the orthogonal group action of a non-degenerate anisotropic quadratic form, we prove that the Legendre symbol is essentially an orbit invariant of level sets under projective equivalence. Next, we demonstrate that the algebraic essence of the Cipolla algorithm is a reduced-order exponential map induced by the Frobenius automorphism, and we unify the computational paths for residue and non-residue classes by exploiting the surjectivity of the field norm. Furthermore, we generalize this framework to arbitrary n-th degree cyclic extensions, proving that both the generalized Cipolla algorithm and the Adleman–Manders–Miller (AMM) algorithm are equivalent to an exponential map determined by a structural constant on the high-dimensional norm torus. Finally, through coordinate transformations of the affine space, the solution of a general quadratic congruence equation is decomposed into a composition of orthogonal group actions and translations. This study dissolves the logical branching of traditional algorithms through the deep structure of algebraic groups, transforming the a priori arithmetic determination into an a posteriori dimensional observation at the endpoint of manifold evolution, and thereby providing an intrinsic geometric interpretation for the computation of higher-degree residues.