Let $R \colon= K[[X]]$ be a formal power series ring over a field $K$. We consider a monic polynomial $f(T) \in R[T]$ as follows$\colon$

$$ f(T) = T^e + c_{e-1}T^{e-1} + \ldots + c_1T + c_0. $$

Let $\overline{f(T)} \in K[T]$ be the reduction of $f(T)$ by the ideal $(X)$. Suppose that both $f(T)$ and $\overline{f(T)}$ are irreducible. Let \begin{align*} & S \colon= R[T]/(f(T)) \\ & L \colon= K[T]/(\overline{f(T)}) \end{align*} be finite extensions of $R$ and $K$, respectively.

Choose an element $\alpha \in L$ and consider ${\mathrm{Norm}}(\alpha) \in K$, where ${\mathrm{Norm}}(\alpha)$ is defined to be the constant term of the minimal polynomial of $\alpha$ over $K$. For example, ${\mathrm{Norm}}(\alpha) = \alpha$ for $\alpha \in K$.

## Q. Choose an element $\beta \in R$ such that $\beta \mapsto {\mathrm{Norm}}(\alpha)$ via the surjection $R\twoheadrightarrow K$ by ${\mathrm{mod}}\,(X)$. Does there always exist a lift $\gamma \in S$ of $\alpha \in L$ such that the following two conditions are satisfied?$\colon$

\begin{align*} \gamma & \equiv \alpha \quad {\mathrm{mod}}\,(X) \\ {\mathrm{Norm}}(\gamma) & = \beta, \end{align*}