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Let $B/A$ be an extension of discrete valuation rings such that the purely inseparable extension of residue fields $l/k$ has a primitive element, i.e., $l=k(y)$ for some element $y$ in $l$. I want to know if one can plug in another discrete valuation ring as follows.

Let $x$ be a lift of $y$ to $B$. Is $A[x]$ a dvr? What are necessary and sufficient conditions?

Of course, the answer is yes if $e=1$. Then $A[x] = B$.

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    I presume you mean "let $x$ be a lift of $y$ to $B$"?2012-01-28

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As $k\subseteq l$ is a purely inseparable extension, characteristic of $k$ should be positive, say $p$. The only fields I know with positive characteristic are finite fields and $F(T_1,\cdots,T_n)$ (where $F$ is a field with positive characteristic like finite fields). I started to make an example.

Let $K=F_{p^n}[T]$ and $L=F_{p^m}[T]$, $n$ x) then $A[x]=B$).

Now if $A$ be trivial then since $T,T^{-1}\in A\subseteq B$, $B$ should be trivial which we said it is not the case.

Can we guess this result is global, I mean the answer of the main question should be "The necessary and sufficient condition is $y\in A[x]$"?

If we change the question in this form that instead of the residue fields, $K\subseteq L$ is a purely extension then except the above example we have another class of examples with the same result. We can get $K=F_{p^n},L=F_{p^m}$, $n

Again can we say answer in the new case is "The necessary and sufficient condition is $y\in A[x]$"?

Here at least I gave an answer in a class of examples.