Definition 10.153.1. Let $(R, \mathfrak m, \kappa )$ be a local ring.

1. We say $R$ is henselian if for every monic $f \in R[T]$ and every root $a_0 \in \kappa$ of $\overline{f}$ such that $\overline{f'}(a_0) \not= 0$ there exists an $a \in R$ such that $f(a) = 0$ and $a_0 = \overline{a}$.

2. We say $R$ is strictly henselian if $R$ is henselian and its residue field is separably algebraically closed.

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