Canonical Sections of Artin–Schreier–Witt Extensions and an Algebraic Expression of the Reciprocity Map in Local Fields of Characteristic p

Let K be a local field of characteristic p. We present a systematic study of the explicit expression of the local reciprocity map for totally ramified cyclic extensions of K of degree pᵉ. Classical Artin–Schreier–Witt theory provides generators of the extension only by implicit existence, which forces the computation of the reciprocity law to resort to transcendental differential residue operators and prevents a purely algebraic closure. Introducing the **canonical section operator** L as a new tool, we impose the canonical constraint that the constant term of the Puiseux series vanish on solutions of the Witt equation, thereby completely eliminating the multi-valued ambiguity of the root-finding process. On this foundation, we establish and prove a **layerwise AS decoupling theorem**, which precisely reduces the solution of a higher-order Witt equation to a finite-step Fₚ-linear recurrence. We further prove that the canonical generator strictly matches the ramification filtration, and that the Galois action can be read out by a direct algebraic extraction of constant terms. We prove that the trace map is algebraically expandable on the Artin–Schreier tower, whereby the expression of the reciprocity translation vector is equivalently transformed from a transcendental form depending on an external residue operator into an intrinsic algebraic constraint on the generator. We thereby establish an **additive-bridging, purely algebraic expression paradigm** for local class field theory in characteristic p, providing a new algebraic foundation for the further algorithmization of explicit reciprocity laws.