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Mathlib.NumberTheory.Padics.Hensel

Hensel's lemma on ℤ_p #

This file proves Hensel's lemma on ℤ_p, roughly following Keith Conrad's writeup: http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/hensel.pdf

Hensel's lemma gives a simple condition for the existence of a root of a polynomial.

The proof and motivation are described in the paper R. Y. Lewis, A formal proof of Hensel's lemma over the p-adic integers.

References #

Tags #

p-adic, p adic, padic, p-adic integer

theorem limit_zero_of_norm_tendsto_zero {p : } [Fact (Nat.Prime p)] {ncs : CauSeq ℤ_[p] norm} {F : Polynomial ℤ_[p]} (hnorm : Filter.Tendsto (fun (i : ) => Polynomial.eval (ncs i) F) Filter.atTop (nhds 0)) :
Polynomial.eval ncs.lim F = 0
theorem hensels_lemma {p : } [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (hnorm : Polynomial.eval a F < Polynomial.eval a (Polynomial.derivative F) ^ 2) :
∃ (z : ℤ_[p]), Polynomial.eval z F = 0 z - a < Polynomial.eval a (Polynomial.derivative F) Polynomial.eval z (Polynomial.derivative F) = Polynomial.eval a (Polynomial.derivative F) ∀ (z' : ℤ_[p]), Polynomial.eval z' F = 0z' - a < Polynomial.eval a (Polynomial.derivative F)z' = z