Hensel's lemma

For a complete discrete valuation ring $R$, a simple root of a polynomial modulo the maximal ideal lifts uniquely to a root in $R$.

For $n\in {\mathbb{Z}}_3$, the equation $x^3+3x+y^3+3y=n$ has a solution in the p-adic integers exactly for the displayed residue classes. The polynomial $F(t)=t^3+3t$ has values $0,4,5$ modulo nine, whose pairwise sums are $0,1,4,5,8$. For sufficiency choose initial pairs $(0,0),(-1,-1),(1,0),(-1,0),(1,1)$ for those five classes respectively, fix the second coordinate, and lift the first. Its derivative is $3(x^2+1)$, always of 3-adic norm $1/3$, while its second Taylor coefficient is $3x$, of norm below one. The residual is divisible by nine, so either the strict or the Boundary case of Hensel's lemma with a small quadratic coefficient applies.

A monic polynomial over a complete discrete valuation ring factors with prescribed coprime monic residue factors. Given monic lifts $g_r,h_r$ correct modulo ${\pi }^r$, the degree-bounded map $(u,v)\mapsto u{\bar{h}}_r+v{\bar{g}}_r$ is a bijection over the residue field, by coprimality and equality of dimensions. Solve it for the error $(f-g_r{h}_r)/{\pi }^r$, and replace the lifts by $g_r+{\pi }^{r}u,h_r+{\pi }^{r}v$. Their coefficients converge by completeness and the limiting product is $f$. The completeness assumption cannot be replaced just by localization.

For a complete discrete valuation ring $R$, let $f(a_1)\in (\pi )$ and let $f^{\prime }(a_1)$ be a unit. If $f(a_n)\in ({\pi }^n)$, choose $c$ modulo $\pi$ solving $f(a_n)/{\pi }^n+c{f}^{\prime }(a_n)\equiv 0$ and put $a_{n+1}=a_n+c{\pi }^n$. Taylor expansion gives $f(a_{n+1})\in ({\pi }^{n+1})$. The sequence is Cauchy and its limit is a root. If two roots lift the same residue, factoring $f(x)-f(y)$ leaves $x-y$ times a unit, proving uniqueness.

Suppose a Non-Archimedean absolute value on $K$ has a unique extension of an absolute value to every finite extension. Factor a monic $g\in {\mathcal{O}}_K[X]$ into monic irreducible factors over $K$. All factors have integral coefficients because their roots have absolute value at most one. By pure-power reduction of a monic irreducible polynomial, each factor reduces to a power of a single irreducible polynomial. If $\overline{g}={\overline{g}}_1{\overline{g}}_2$ is a coprime monic factorization, assign each whole irreducible factor of $g$, with its multiplicity, to the unique side containing its residue factor. The resulting monic $g_1,g_2\in {\mathcal{O}}_K[X]$ satisfy $g=g_1{g}_2$ and reduce to the prescribed factors. Completeness of $K$ is not required for this proof.

For a non-Archimedean local field $K$, an element of $K^{\times }$ is an $m$th power for infinitely many positive integers $m$ exactly when it belongs to ${\mathcal{O}}_K^{\times }$. Valuations prove necessity; for sufficiency, choose infinitely many $m$ coprime to the residue characteristic and to $|k^{\times }|$, for which exponentiation by $m$ is an automorphism of the unit group.

The polynomial $X^4+9X^2-2$ has three irreducible factors over ${\mathbb{Q}}_2$, of degrees one, one, and two. Its quadratic in $X^2$ has one unit root congruent to one modulo eight, hence a square, and one root of valuation one, whose associated quadratic factor is Eisenstein.

Let $K$ be complete for a discrete valuation $v$. If $f\in {\mathcal{O}}_K[X]$ andthen Newton iteration over a valued field converges to a root $\alpha \in {\mathcal{O}}_K$ with $v(\alpha -a)>v(f^{\prime }(a))$.

Let $R$ be a complete discrete valuation ring, $f\in R[X]$, and $d=|f^{\prime }(a)|$ satisfy $0<d<1$ and $|f(a)|=d^2$. Suppose the second Taylor coefficient of $f$ at every point of $R$ has norm below one. One Newton iteration over a valued field step has correction $h=-f(a)/f^{\prime }(a)$ of norm $d$. Its residual has quadratic term of norm strictly below $d^2$, and all higher terms have norm at most $d^3<d^2$. The derivative changes by a term of norm below $d$, so its norm remains $d$. The strong form of Hensel lemma therefore applies from the new point, producing a root within distance $d$ of the original point. The second Taylor coefficient is $f^{\prime \prime }/2$ when that quotient is meaningful, or the division-free binomial-coefficient Taylor expression in arbitrary characteristic. A Newton sequence that reaches a root is thereafter constant; residuals decrease strictly only before that event.

Starting from $a_0$, Newton iteration sets $a_{n+1}=a_n-f(a_n)/f^{\prime }(a_n)$. Under the strong form of Hensel lemma inequality, the correction valuations grow at least geometrically, so $(a_n)$ is a Cauchy sequence converging to a root.