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

Abstract

Let v be a henselian valuation of arbitrary rank of a field K and v ̃ 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 whenever v ̃ ( 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.