Hensel's Lemma for the p-adic Integers


Title: Hensel's Lemma for the p-adic Integers
Author: Aaron Crighton (crightoa /at/ mcmaster /dot/ ca)
Submission date: 2021-03-23
Abstract: We formalize the ring of p-adic integers within the framework of the HOL-Algebra library. The carrier of the ring is formalized as the inverse limit of quotients of the integers by powers of a fixed prime p. We define an integer-valued valuation, as well as an extended-integer valued valuation which sends 0 to the infinite element. Basic topological facts about the p-adic integers are formalized, including completeness and sequential compactness. Taylor expansions of polynomials over a commutative ring are defined, culminating in the formalization of Hensel's Lemma based on a proof due to Keith Conrad.
  author  = {Aaron Crighton},
  title   = {Hensel's Lemma for the p-adic Integers},
  journal = {Archive of Formal Proofs},
  month   = mar,
  year    = 2021,
  note    = {\url{https://isa-afp.org/entries/Padic_Ints.html},
            Formal proof development},
  ISSN    = {2150-914x},
License: BSD License
Status: [ok] This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.