A formal proof of Hensel's lemma over the p-adic integers

The field of $p$-adic numbers $\mathbb{Q}_p$ and the ring of $p$-adic integers $\mathbb{Z}_p$ are essential constructions of modern number theory. Hensel's lemma, described by Gouv\^ea as the "most important algebraic property of the $p$-adic numbers," shows the existence of roots of polynomials over $\mathbb{Z}_p$ provided an initial seed point. The theorem can be proved for the $p$-adics with significantly weaker hypotheses than for general rings. We construct $\mathbb{Q}_p$ and $\mathbb{Z}_p$ in the Lean proof assistant, with various associated algebraic properties, and formally prove a strong form of Hensel's lemma. The proof lies at the intersection of algebraic and analytic reasoning and demonstrates how the Lean mathematical library handles such a heterogeneous topic.Comment: CPP 201

A formal proof of Hensel's lemma over the p-adic integers

The field of p-adic numbers p and the ring of p-adic integers p are essential constructions of modern number theory. Hensels lemma, described by Gouva as the most important algebraic property of the p-adic numbers, shows the existence of roots of polynomials over p provided an initial seed point. The theorem can be proved for the p-adics with significantly weaker hypotheses than for general rings. We construct p and p in the Lean proof assistant, with various associated algebraic properties, and formally prove a strong form of Hensels lemma. The proof lies at the intersection of algebraic and analytic reasoning and demonstrates how the Lean mathematical library handles such a heterogeneous topic