On Brown’s constant associated with irreducible polynomials over henselian valued fields

Abstract

a b s t r a c tLet v be a henselian valuation of arbitrary rank of a field K andṽ be the prolongation of v to the algebraic closure K of K with value group G. In 2008, Ron Brown gave a class P of monic irreducible polynomials over K such that to each g(x) belonging to P , there corresponds a smallest constant λ g belonging to G (referred to as Brown's constant) with the property that wheneverṽ(g(β)) is more than λ g with K (β) a tamely ramified extension of (K , v), then K (β) contains a root of g(x). In this paper, we determine explicitly this constant besides giving an important property of λ g without assuming that K (β)/K is tamely ramified.