Let us consider a formal power series ring $A_n \colon= K[[X_1,\ldots,X_n]]$ with $0 \ll n < \infty$ and we shall consider a prime ideal ${\frak P}$ of $A_n$ such that $1 < {\mathrm{ht}}({\frak P}) < n$.

We shall choose an irreducible element $f \in {\frak P}$ such that up to a variables transformation and by abuse of notations we have

\begin{equation*}
f = a_e + a_{e-1}X_1 + \ldots + a_{1}(X_1)^{e-1} + X_1^e,
\end{equation*}
where $a_1,\ldots,a_e \in K[[X_2,\ldots,X_n]]$.

The ring $R_n \colon= A_n/f$ is finite over the ring $A_{n-1} \colon= K[[X_2,\ldots,X_n]]$. Let us denote by ${\frak Q}$ the unique prime of $A_{n-1}$ lying below ${\frak P}$.

## Q. Can I always find $f$ such that ${\frak P}$ is étale over ${\frak Q}$? Or equivalently for the discriminant $\Delta_f$ of $f$, can I have $\Delta_f \notin {\frak Q}$ ?