Chapter 16: Hensel Lifting and Local Fields

The CRT Decomposition Theorem

split the primorial residue ring ℤ / M_d ℤ into a product of prime-power components. This chapter climbs those components vertically: given a root modulo p, we lift it constructively to a root modulo p², then p³, and so on without end. The mechanism is modular Newton iteration—a correction-by-divisibility procedure that requires no signed arithmetic and no convergence analysis. The lifting is unique at each stage (a p-adic contraction in the classical language; a tower-coherent singleton in ours). Assembling all stages produces ℤ_p^τ = lim← ℤ / p^n ℤ, the τ-native p-adic integers—an inverse limit that lives inside Category τ with a canonical NF address. The p-adic valuation v_p is the D-coordinate restricted to the p-primary component of the primorial tower. We prove that ℤ_p^τ is “complete” in the τ-sense: every tower-coherent sequence has a unique limit. This is not metric completeness but Global Hartogs (I.T31) restricted to the p-primary sub-tower—no metric, no Cauchy sequences, just tower coherence. The chapter closes with the local-global bridge: the τ-native local fields assemble via CRT into the profinite completion ℤ_τ, laying the foundation for the adelic embedding of Chapter 17.