Irreducible factors of a polynomial and extensions of valuations (Q6610529)

Let \((K, v)\) be an arbitrary rank valued field, let \(\Gamma\) its value group, \(\mathbb F_v\) its residue field, \((K^h, v^h)\) a fixed Henselization, \(R_{v^h}\) the valuation ring of \((K^h, v^h)\), and \(M_{v^h}\) its maximal ideal. For a polynomial \(F(x) \in R_v[x]\), which is congruent to a positive power of a polynomial \(\phi\), where \(\phi \in R_v[x]\) is a monic polynomial, whose reduction \(\overline{\phi(x)}\) is irreducible over \(\mathbb F_v[x]\), let \(F (x) = \phi^n(x) + a_{n-1}(x)\phi^{n-1}(x) +\cdots+ a_0(x)\) be the \(\phi\)-expansion of \(F(x)\) and \(N_\phi(F)\) the \(\phi\)-Newton polygon of \(F(x)\) with respect to \(v\). In this article, the theorem precisely stated below gives a natural generalization of the results in [\textit{A. Jakhar}, Bull. Lond. Math. Soc. 52, No. 1, 158--160 (2020; Zbl 1455.11144)] together with a totally different proof.\N\NTheorem 1. Let \(F(x)\in R_{v^h}[x]\) be a monic polynomial such that \(\overline {F(x)}\) is a power of \(\overline \phi \in \mathbb F_v[x]\) for some monic polynomial \(\phi\in R_{v^h}[x]\), where \(\overline{\phi}\) is irreducible in \(\mathbb F_v[x]\), and \(N_{\phi}(F)=S\) has a single side of slope \(-\lambda\). Let \(e\) be the smallest positive integer satisfying \(e\lambda\in \Gamma\) and \(m=\deg(\phi)\). Then \(F(x)\) has at most \(\frac{n}{m\cdot e}\) monic irreducible factors over \(K^h\) and each irreducible factor is of degree at least \(m\cdot e\), where \(n\) is the degree of \(F(x)\). \N\NFurther in the article, Eisenstein-Schönemann irreducibility criterion is generalized as follows.\N\NTheorem 2. Let \(F(x)\in R_{v^h}[x]\) be a monic polynomial such that \(\overline {F(x)}\) is a power of \(\overline \phi \in \mathbb F_v[x]\) for some monic polynomial \(\phi\in R_{v^h}[x]\), where \(\overline{\phi}\) is irreducible in \(\mathbb F_v[x]\), and \(N_{\phi}(F)=S\) has a single side of slope \(-\lambda\in \mathbb Q\Gamma\). If \(R_\lambda(F)\) is irreducible in \(\mathbb F_\phi[y]\), then \(F(x)\) is irreducible over \(K^h\).\N\NIn this article it is proved that if \(L=K(\alpha)\) is a simple extension of \(K\) generated by a root \(\alpha \in \overline K\) of a monic irreducible polynomial \(F(x)\in R_v[x]\), then there are at most \(r=\frac{n}{e\cdot m}\) distinct valuations \(\omega_i, 1\leq i\leq r\) of \(L\) extending \(v\) (see Theorem 5).\N\NThe number of irreducible factors of \(F(x)\) over \(K^h\) and a bound on the number of distinct valuations of \(L\) extending \(v\) are explained (see Theorem 6 and Corollary 1). Further the examples are vivid.\N\NFor the entire collection see [Zbl 1539.11005].