Norm Folding and Explicit Root Finding for Artin–Schreier Equations over Finite Fields

Addressing the long-standing reliance on a priori trace determination and the absence of explicit unified root-finding formulas for Artin–Schreier equations (x) = Δ over finite fields, this paper reveals the profound algebraic asymmetry with Kummer extensions and establishes a strictly dual unified explicit root-finding theory. We show that under the additive translation of the Galois group, the field norm undergoes an "orthogonal collapse" into the -map (Theorem 1), forming a perfect duality with the "expansion" of the field norm in Kummer extensions. Based on this folding mechanism, we establish an unconditional -homomorphism bridging identity (Theorem 2), and by introducing the trace scalar τ as a linear compensator, we derive the first unified explicit root-finding formula x = LF(z) + τω (Theorem 3) parallel to the Kummer paradigm for characteristic p. This paradigm internalizes the traditional a priori discrete determination into the logic of algebraic operations, achieving an intrinsic unity of "determination as construction," and realizes a dimensional reduction from nonlinear exponent extraction to linear operator extraction in computational complexity.