The Stacks project

Definition 59.32.2. (See Algebra, Definition 10.153.1.) A local ring $(R, \mathfrak m, \kappa )$ is called henselian if for all $f \in R[T]$ monic, for all $a_0 \in \kappa$ such that $\bar f(a_0) = 0$ and $\bar f'(a_0) \neq 0$, there exists an $a \in R$ such that $f(a) = 0$ and $a \bmod \mathfrak m = a_0$.