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 ν̅ 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)there corresponds a smallest constant λ_g belonging to G^~(referred to as Brown’s constant) with the property that whenever ν̅(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.