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 ν(d)>pν(p)/p-1 then (1 + d)^1/P ε 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 ε K then 1 + d is a pth power in K.