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If K_0 is the fraction field of the Witt vectors over an algebraically closedfield k of characteristic p, we calculate upper bounds on the conductor ofhigher ramification for (the Galois closure of) extensions K_0(zeta_{p^r},sqrt[p^r]{a})/K_0, where a is in K_0(zeta_{p^r}). Here zeta_{p^r} is aprimitive p^r-th root of unity. In certain cases, including when a is in K_0and p=2, we calculate the conductor exactly. These calculations can be used todetermine the discriminants of various extensions of Q obtained by adjoiningroots of unity and radicals.

Conductors of wild extensions of local fields, especially in mixed characteristic (0,2)