Hensel's lemma

In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a polynomial equation has a simple root modulo a prime number p, then this root corresponds to a unique root of the same equation modulo any higher power of p, which can be found by iteratively "lifting" the solution modulo successive powers of p. More generally it is used as a generic name for analogues for complete commutative rings (including p-adic fields in particular) of the Newton method for solving equations. Since p-adic analysis is in some ways simpler than real analysis, there are relatively neat criteria guaranteeing a root of a polynomial.

Many equivalent statements of Hensel's lemma exist. Arguably the most common statement is the following.

Assume ${\displaystyle K}$ is a field complete with respect to a normalised discrete valuation ${\displaystyle v}$. Suppose, furthermore, that ${\displaystyle {\mathcal {O}}_{K}}$ is the ring of integers of ${\displaystyle K}$ (i.e. all elements of ${\displaystyle K}$ with non-negative valuation), let ${\displaystyle \pi \in K}$ be such that ${\displaystyle v(\pi )=1}$ and let ${\displaystyle k={\mathcal {O}}_{K}/\pi }$ denote the residue field. Let ${\displaystyle f(X)\in {\mathcal {O}}_{K}[X]}$ be a polynomial with coefficients in ${\displaystyle {\mathcal {O}}_{K}}$. If the reduction ${\displaystyle {\overline {f}}(X)\in k[X]}$ has a simple root (i.e. there exists ${\displaystyle k_{0}\in k}$ such that ${\displaystyle {\overline {f}}(k_{0})=0}$ and ${\displaystyle {\overline {f'}}(k_{0})\neq 0}$), then there exists an unique ${\displaystyle a\in {\mathcal {O}}_{K}}$ such that ${\displaystyle f(a)=0}$ and the reduction ${\displaystyle {\overline {a}}=k_{0}}$ in ${\displaystyle k}$.

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