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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