(Hensel's lemma)
Let \((R, \mfm, k)\) be a complete Noetherian local ring and \(f \in R[x]\) monic. If the reduction \(\bar{f} \in k[x]\) factors as \(\bar{f} = g_0 h_0\) with \(g_0, h_0\) monic and coprime, then \(f = g h\) with \(g, h \in R[x]\) monic, \(\bar{g} = g_0\) and \(\bar{h} = h_0\). In particular a simple root of \(\bar{f}\) in \(k\) lifts to a root of \(f\) in \(R\).
(Henselian ring and henselization)
A local ring \((R, \mfm)\) is henselian if the conclusion of Hensel’s lemma holds for every monic \(f \in R[x]\). The henselization \(R^h\) of a local ring \(R\) is a henselian local ring with a local homomorphism \(R \to R^h\) through which every local homomorphism from \(R\) to a henselian local ring factors uniquely. It is the colimit of the local rings \(\OO_{U,u}\) over étale neighbourhoods \((U, u) \to (\Spec R, \mfm)\) with trivial residue field extension.
Complete local rings are henselian, and \(R \to R^h \to \hat{R}\). For \(R = k[x]_{(x)}\), \(R^h\) is the ring of power series in \(k[[x]]\) that are algebraic over \(k(x)\); it contains \(\sqrt{1+x}\) in characteristic not \(2\), which \(R\) does not, and it does not contain \(e^x\) when \(k = \CC\).