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

Let v be a henselian valuation of arbitrary rank of a field K and be the prolongation of v to the algebraic closure of K with value group . 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 λg belonging to (referred to as Brown’s constant) with the property that whenever is more than λg 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 λg without assuming that K(β)/K is tamely ramified.