Remark

The corollary applies notably when $e$ is invertible in $\mathbf{A}$; we can then also deduce the existence of $b$ from Hensel's Theorem, for the canonical image of $f(\mathbf{X})$ in $(\mathbf{A}/\mathfrak{m})\{\mathbf{X}\}$ is of the form $(\mathbf{X} - a)f_1(\mathbf{X}), \mathbf{X} - a$ and $f_1(\mathbf{X})$ being strongly relatively prime, for $f_1(\alpha) = f'(\alpha)$ is the image of $e$ (no. 1, Example).

Examples (1) Let $p$ be a prime number $\neq 2$ and $n$ an integer whose class mod. $p$ is a square $\neq 0$ in the prime field $\mathbf{F}_p$. If $\mathbf{Z}_p$ is the ring of $p$-adic integers (§ 2, no. 12, Example 3), the application of Corollary 1 to the polynomial $\mathbf{X}^2 - n$ shows that $n$ is a square in $\mathbf{Z}_p$; for example 7 is a square in $\mathbf{Z}_3$. (2) Let $\mathbf{A} = \mathbf{K}[[\mathbf{Y}]]$ be the ring of formal power series in one indeterminate with coefficients in a commutative field $\mathbf{K}$; with the $(\mathbf{Y})$-adic topology, the ring $\mathbf{A}$ is Hausdorff and complete (§ 2, no. 6, Corollary to Proposition 6) and the mapping $f(\mathbf{Y}) \mapsto f(0)$ defines by passing to the quotient ring an isomorphism of $\mathbf{A}/(\mathbf{Y})$ onto the field $\mathbf{K}$. By Corollary 1, if $F(\mathbf{Y}, \mathbf{X})$ is a polynomial in $\mathbf{X}$ with coefficients in $\mathbf{A}$ and $a$ is a simple root of $F(0, \mathbf{X})$ in $\mathbf{K}$, there exists a unique formal power series $f(\mathbf{Y})$ such that $f(0) = a$ and $F(\mathbf{Y}, f(\mathbf{Y})) = 0$.