A regular system of parameters of a ramified complete regular local ring Let $(R,\mathfrak m)$ be a ramified  complete regular local ring of dimension $n$ with $\operatorname{char} R/\mathfrak m=p$. Then $\operatorname{ht}(pR)=1$ and the question is:  

How to choose a regular system of parameters $x_1,\dots,x_n$ of $R$ such that $p,x_2,\dots,x_n$ is a system of parameters of $R$, that is, $\sqrt{(p,x_2,\dots,x_n)}=\mathfrak m$?

 A: Let $x_1,\dots,x_n$ be a regular system of parameters, that is, $\mathfrak m=(x_1,\dots,x_n)$. Then $\overline R=R/(x_1,\dots,x_{n-1})$ is a regular local ring of dimension $1$ and its maximal ideal $\overline{\mathfrak m}=\mathfrak m/(x_1,\dots,x_{n-1})$ is generated by $\overline x_n$. We have $\overline p\in\overline{\mathfrak m}^2$, and therefore there exists an invertible element $\overline a\in\overline R$ and $t\geq 2$ such that $\overline p=\overline a\overline x_n^t$. This shows that $x_n^t\in(p,x_1,\dots,x_{n-1})$ and thus we get $\mathfrak m^t\subseteq (p,x_1,\dots,x_{n-1})$ which means that $p,x_1,\dots,x_{n-1}$ is a system of parameters.
However a problem can occur in the proof above: we could have $\overline p=\overline 0$, that is, $p\in(x_1,\dots,x_{n-1})$. 
Now our task is to choose a regular system of parameters $x_1,\dots,x_n$ such that $p\notin(x_1,\dots,x_{n-1})$. We shall do this by induction on $n\ge 1$. If $n=1$ there is nothing to prove. In the general case take $(y_1,\dots,y_n)=\mathfrak m$ an arbitrary regular system of parameters. We know that $\operatorname{ht}(p)=1$ and consider $\mathfrak p_1,\dots,\mathfrak p_r$ all height $1$ prime ideals of $R$ containing $(p)$. None of these prime ideals contain all $y_i$. We can assume $y_n\notin\mathfrak p_1,\dots,\mathfrak p_s$ and $y_n\in\mathfrak p_{s+1},\dots,\mathfrak p_r$ and define $x_n=y_n+\sum_{j=s+1}^ra_jy_{k_j}$, where $y_{k_j}\notin\mathfrak p_j$ and $a_i\in\bigcap_{j\neq i}\mathfrak p_j-\mathfrak p_i$. Then $x_n\notin\bigcup_{i=1}^r\mathfrak p_i$. The elements $y_1,\dots,y_{n-1},x_n$ are also a regular system of parameters and moreover $p\notin (x_n)$, otherwise $(p)\subseteq (x_n)$ and $(x_n)$ is a prime ideal of height $1$, so $(x_n)$ must coincide with some $\mathfrak p_i$, a contradiction. All we have to do now is to reduce the problem to $R/(x_n)$ and apply the induction hypothesis.
