# Étale fibration for $K[[X_1,...,X_n]]$

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 $$\Delta_f \notin {\frak Q}$$ ?

A general ideal $$\mathfrak P$$ need not contain an element $$f$$ that is a polynomial in one of the variables. The existence of such an element $$f$$ is a strong restriction on the ideal $$\mathfrak P$$. An even stronger restriction is requiring, in addition, that the polynomial $$f$$ be monic.
If we take an ideal $$\mathfrak P$$ containing a monic polynomial $$f$$ in $$X_1$$ then the answer to your question is positive if $$K$$ is of characteristic 0 and negative if $$K$$ is of characteristic $$p>0$$. Namely, if $$char\ K=0$$ and $$\mathfrak P$$ is not étale over $$\mathfrak Q$$, replace $$f$$ by $$\frac{\partial f}{\partial X_1}$$ and keep repeating this process as many times as necessary.
If $$char\ K=p>0$$, take $$n=3$$ (we need $$n\ge3$$ in order the inequalities on $$ht\ \mathfrak P$$ to be respected). Let $$\mathfrak P$$ be an ideal generated by $$X_1^p-g$$ and $$h$$, where $$g$$ and $$h$$ are sufficiently general power series in $$X_2$$ and $$X_3$$ without constant or linear terms. Then $$\mathfrak P$$ contains a monic polynomial, but none that make $$\mathfrak P$$ étale over $$\mathfrak Q$$.