This paper aims to establish an algebraic-geometric framework for solving the $n$-th degree congruence equation $x^n \equiv \Delta \pmod p$ over finite fields $\mathbb{F}_p$, transforming the traditional arithmetic determination of residue classes into the intersection problem of algebraic varieties and the base field. On the algebraic torus $R_{\mathbb{F}_{p^2}/\mathbb{F}_p}(\mathbb{G}_m)$ of the quadratic extension, utilizing the action of the orthogonal group on non-degenerate anisotropic quadratic forms, it is proved that the Legendre symbol is essentially the orbit invariant of the level sets under projective equivalence. It is demonstrated that the algebraic essence of Cipolla's algorithm is a half-period exponential map induced by the Frobenius automorphism, and the computational paths for residues and non-residues are unified by utilizing the surjectivity of the field norm. We extend this framework to general cyclic extensions of degree $n$, proving that both the generalized Cipolla and the Adleman-Manders-Miller (AMM) algorithms are equivalent to the exponential map determined by structure constants on higher-dimensional norm tori. Through coordinate transformations in affine space, the solution of the general quadratic congruence equation is decomposed into the composition of an orthogonal group action and a translation transformation. This study eliminates the logical branching of traditional algorithms from the deep structure of algebraic groups, reducing the a priori arithmetic determination to an a posteriori observation of dimensionality at the evolution endpoint on the manifold, thereby providing an intrinsic geometric interpretation for the computation of higher-degree residues.