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 xⁿ ≡ Δ p over a finite field Fₚ 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 Δ an n-th degree extension field L = Fₚ(ω) can be dynamically constructed in a uniform manner, and establishes the norm identity N(t-ω) = Δ within this field. Chapter 2 carries out a Galois symmetry analysis on the dynamically constructed extension, giving the explicit formula φ(∑ aᵢ ωⁱ) = ∑ aᵢ ζⁱ ωⁱ for the Frobenius automorphism (where ζ 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-ω)(pⁿ-1)/n(p-1) satisfies rⁿ = Δ; 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 Δ to the random parameter tⁿ-Δ, the core computation path executes the same algebraic operation sequence for all Δ, and the residuosity of Δ 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.