Suppose that $R$ is a Noetherian complete domain over a field $K$.

Suppose that a monic polynomial $f(X) \in R[X]$ (i.e., the highest degree $X^e$ in $f$ has the coefficient $1$), satisfies the following two conditions$\colon$

$R[X]/(f(X))$ is *not* integral.

$f(X) = g(X)^l$ in $F(R)[X]$ for some integer $l > 1$, where $g(X)$ is an irreducible polynomial in $F(R)[X]$ and $F(R)$ is the fractional field of $R$.

## Q. Then, does the following equality hold for some $G(X) \in R[X]$$\colon$ \begin{equation*}
f(X) = G(X)^l~?
\end{equation*}