A characterization of Krasner's Constant

Abstract

Let v be a henselian valuation of any rank of a eld K with value group G and υ^- be its unique prolongation to a xed algebraic closure K^- of K: For an element α of K^-\K; which is separable over K; let ωK (α) denote the well known Krasner's constant given by max{υ^-(α - α′)│α′ ≠ α runs over K conjugates of α. In 1946, Krasner proved that if β belonging to K^- is such that υ^-(α - β ) > ωK (α); then K(α) ⊆ K(β): In this paper, we investigate whether ωK (α) is the smallest among all the elements λ of the divisible closure of G which have the property that whenever υ^-(α -β) > λ, β ∈ K^-, then K(α) ⊆ K(β).