On Brown's constant associated with irreducible polynomials over Henselian valued fields
Let \(v\) be a Henselian valuation of an arbitrary rank of a field \(K\) and \(\tilde v\) is the unique prolongation of \(v\) to the algebraic closure \(\tilde K\) of \(K\) with value group \(\tilde G\). In 2008, Brown gave a class of monic irreducible polynomials over \(K\) satisfying the following property: To every \(g(X)\in{\mathcal P}\), one can associate a constant \(\lambda_g\in\tilde G\) such that whenever \(K(\beta)\) is a tamely ramified extension of \((K,v)\), \(\beta\in\tilde K\) and \(\tilde v(g(\beta))>\lambda_g\), then \(K(\beta)\) contains a root of the polynomial \(g(X)\). Moreover, the constant \(\lambda_g\) is the smallest with the above property. This constant is referred to as Brown's constant. In this paper, the author determines explicitly Brown's constant for all possible irreducible polynomials \(g(X)\) and shows that this constant satisfies an important property even without the assumption that \(K(\beta)/K\) is tamely ramified. She shows that this constant can be associated to any monic irreducible polynomial \(g(X)\in K[X]\) provided \(K(\theta)\) is a defectless extension of \((K,v)\) where \(\theta\) is a root of \(g(X)\).
scientific article; zbMATH DE number 5775718