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

Abstract:

If $K_0$ is the fraction field of the Witt vectors over an algebraically closed field $k$ of characteristic $p$, we calculate upper bounds on the conductor of higher ramification for (the Galois closure of) extensions of the form $K_0(\zeta _{p^c}, \sqrt [p^c]{a})/K_0$, where $a \in K_0(\zeta _{p^c})$. Here $\zeta _{p^c}$ is a primitive $p^c$th root of unity. In certain cases, including when $a \in K_0$ and $p=2$, we calculate the conductor exactly. These calculations can be used to determine the discriminants of various extensions of $\mathbb {Q}$ obtained by adjoining roots of unity and radicals.