On pth Roots in Henselian Valued Fields

Abstract

Let v be a Henselian valuation of arbitrary rank of a field K of characteristic zero with value group G and residue field of characteristic p > 0. Suppose that K contains a primitive pth root of unity. It is well known that if d is an element of K with then (1 + d)1/p ∈ K. In this article we investigate whether is the smallest among all elements λ of G which have the property that whenever v(d) > λ, d ∈ K then 1 + d is a pth power in K.