Hensel's lemma

Last updated January 27, 2022

Suppose $f\in\mathbb{Z}_p[t]$. Let $f'$ be its derivative, and $n$ a positive integer.

Theorem. Suppose $x\in\mathbb{Z}_p$ is such that $v_p(f'(x))=0$ and $f(x)\equiv 0\mod p^n$. Then there exists $y\in\mathbb{Z}_p$ such that

1. $f(y)\equiv 0\mod p^{n+1}$
2. $v_p(f'(y))=0$
3. $y\equiv x\mod p^n$.

This is essentially saying a (simple!) root modulo $p$ can be lifted to a root modulo any higher power of $p$. By taking the limit, it follows that the root modulo $p$ can be lifted to a root over the $p$-adic integers.