Generalizations of Hensel's lemma and the nearest root method (Q695791)

Let \(v\) be a Henselian valuation of arbitrary rank of a field \(K\) and \(\tilde{v}\) be the prolongation of \(v\) to the algebraic closure \(\widetilde{K}\) of \(K\) with value group \(\widetilde{G}\). In 2008, \textit{Ron Brown} [Commun. Algebra 37, No. 7, 2169--2183 (2009; Zbl 1205.12006)] gave a class \(\mathcal{P}\) of monic irreducible polynomials over \(K\) such that to each \(g(x)\) belonging to \(\mathcal{P}\), there corresponds a smallest constant \(\lambda_g\) belonging to \(\widetilde{G}\) with the property that whenever \(\tilde{v}(g(\beta))\) is more than \(\lambda_g\) with \(K(\beta)\) a tamely ramified extension of \((K,v)\), then \(K(\beta)\) contains a root of \(g(x)\). In 2010, a method was described in [the reviewer, J. Pure Appl. Algebra 214, No. 12, 2294--2300 (2010; Zbl 1244.12002)], to explicitly determine this constant besides giving its significant properties. In the present article the author proves a result leading to a new proof of these properties.