# 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. BibTeX: @article{Padic_Ints-AFP, 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