Unified Extension Field Solution of n-th Power Congruences over Finite Fields: Galois Symmetry and the Norm Method

The classical solution of the $n$-th power congruence $x^n \equiv \Delta \pmod p$ over a finite field $\mathbb{F}_p$ takes the $n$-th power residue symbol as its branching criterion, thereby separating the equation into two disjoint algebraic branches: "solvable" and "unsolvable." This paper departs from the theory of algebraic extensions and Galois theory, constructing a unified algebraic solution paradigm in which this rupture is healed within the extension field. Chapter 1 proves, via the binomial irreducibility criterion and character sum estimates, that for any $\Delta$ an $n$-th degree extension field $L = \mathbb{F}_p(\omega)$ can be dynamically constructed in a uniform manner, and establishes the norm identity $\operatorname{N}(t-\omega) = \Delta$ within this field. Chapter 2 carries out a Galois symmetry analysis on the dynamically constructed extension, giving the explicit formula $\phi(\sum a_i \omega^i) = \sum a_i \zeta^i \omega^i$ for the Frobenius automorphism (where $\zeta$ is a primitive $n$-th root of unity), and reveals the algebraic essence of the generalized Cipolla algorithm---$n$-th root extraction is equivalent to taking the $n$-th root of the field norm: $r = (t-\omega)^{(p^n-1)/n(p-1)}$ satisfies $r^n = \Delta$; the orbit structure of the root set under Frobenius action is also precisely characterized. Chapter 3 transforms this algebraic structure into a computational paradigm of branch isolation: the object of the irreducibility test shifts from the input $\Delta$ to the random parameter $t^n-\Delta$, the core computation path executes the same algebraic operation sequence for all $\Delta$, and the residuosity of $\Delta$ is revealed a posteriori by whether the power basis coefficients of the computed result vanish. This paradigm eliminates, on the algebraic-structural level, conditional branching dependent on input data properties, providing structural support for constant-time implementations resistant to side-channel attacks.