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

TL;DR

This paper provides bounds and exact calculations for the ramification conductors of certain wild extensions of local fields, especially in mixed characteristic with p=2, aiding in understanding their discriminants.

Contribution

It introduces new methods to compute upper bounds and exact values of conductors for wild ramification in mixed characteristic local field extensions.

Findings

Calculated upper bounds for conductors in specific wild extensions.

Determined exact conductors when a is in K_0 and p=2.

Applied results to find discriminants of extensions of Q with roots of unity and radicals.

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 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 a primitive p^r-th root of unity. In certain cases, including when a is in K_0 and p=2, we calculate the conductor exactly. These calculations can be used to determine the discriminants of various extensions of Q obtained by adjoining roots of unity and radicals.