Lemma 15.115.11. Let $A$ be a discrete valuation ring with fraction field $K$ of characteristic $p > 0$. Let $\xi \in K$. Let $L$ be an extension of $K$ obtained by adjoining a root of $z^ p - z = \xi $. Then $L/K$ is Galois and one of the following happens

$L = K$,

$L/K$ is unramified with respect to $A$ of degree $p$,

$L/K$ is totally ramified with respect to $A$ with ramification index $p$, and

the integral closure $B$ of $A$ in $L$ is a discrete valuation ring, $A \subset B$ is weakly unramified, and $A \to B$ induces a purely inseparable residue field extension of degree $p$.

Let $\pi $ be a uniformizer of $A$. We have the following implications:

If $\xi \in A$, then we are in case (1) or (2).

If $\xi = \pi ^{-n}a$ where $n > 0$ is not divisible by $p$ and $a$ is a unit in $A$, then we are in case (3)

If $\xi = \pi ^{-n} a$ where $n > 0$ is divisible by $p$ and the image of $a$ in $\kappa _ A$ is not a $p$th power, then we are in case (4).

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