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Lemma 15.115.7. Let $A$ be a discrete valuation ring with uniformizer $\pi $. If the residue characteristic of $A$ is $p > 0$, then for every $n > 1$ and $p$-power $q$ there exists a degree $q$ separable extension $L/K$ totally ramified with respect to $A$ such that the integral closure $B$ of $A$ in $L$ has ramification index $q$ and a uniformizer $\pi _ B$ such that $\pi _ B^ q = \pi + \pi ^ n b$ and $\pi _ B^ q = \pi + (\pi _ B)^{nq}b'$ for some $b, b' \in B$.

Proof. If the characteristic of $K$ is zero, then we can take the extension given by $\pi _ B^ q = \pi $, see Lemma 15.114.2. If the characteristic of $K$ is $p > 0$, then we can take the extension of $K$ given by $z^ q - \pi ^ n z = \pi ^{1 - q}$. Namely, then we see that $y^ q - \pi ^{n + q - 1} y = \pi $ where $y = \pi z$. Taking $\pi _ B = y$ we obtain the desired result. $\square$

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