On the geometric determination of extensions of non-Archimedean absolute values

Year, pages: 2023, 87 - 102

Let $| |$ be a discrete non-archimedean absolute value of a field $K$ with valuation ring $\mathcal{O}$, maximal ideal $\mathcal{M}$ and residue field $\mathbb F=\mathcal{O}/\mathcal{M}$. Let $L$ be a simple finite extension of $K$ generated by a root $α$ of a monic irreducible polynomial $F \in \mathcal{O}[x]$. Assume that $\overline{F}=\overline{φ}^l$ in $\mathbb F[x]$ for some monic polynomial $φ \in \mathcal{O}[x]$ whose reduction modulo $\mathcal{M}$ is irreducible, the $φ$-Newton polygon $N_φ^-(F)$ has a single side of negative slope $λ$, and the residual polynomial $R_λ(F)(y)$ has no multiple factors in $\mathbb F_φ[y]$. In this paper, we describe all absolute values of $L$ extending $| |$. The problem is classical but our approach uses new ideas. Some useful remarks and computational examples are given to highlight some improvements due to our results.

Faris, M., El Fadil, L. 2023. On the geometric determination of extensions of non-Archimedean absolute values. In Tatra Mountains Mathematical Publications, vol. 83, no.1, pp. 87-102. 1210-3195. DOI: https://doi.org/10.2478/tmmp-2023-0007

Faris, M., El Fadil, L. (2023). On the geometric determination of extensions of non-Archimedean absolute values. Tatra Mountains Mathematical Publications, 83(1), 87-102. 1210-3195. DOI: https://doi.org/10.2478/tmmp-2023-0007

Publisher: Mathematical Institute, Slovak Academy of Sciences, Bratislava

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