Absorbent property, Krasner type lemmas and spectral norms for a class of valued fields

Abstract

Let $(K,\varphi)$ be a perfect valued field of rank 1, let $\overline{\varphi}$ be an extension of the absolute (multiplicative) value $\varphi$ to a fixed algebraic closure $\overline{K}$ and let $\| .\|_{\varphi}$ be the corresponding spectral norm on $K$. Let $(\widetilde{\overline{K}},\| .\|_{\varphi}^{\,\tilde{}})$ be a fixed completion of $(\overline{K},\| .\|_{\varphi})$. In this paper we generalize a result of A. Ostrowski~[8] relative to the absorbent property of a subfield, from the case of a complete non-Archimedian valued field of characteristic 0 to our ring $(\widetilde{\overline{K}},\| .\|_{\varphi}^{\,\tilde{}})$ (see Theorem 1, Theorem 4). We also apply these results to discuss in a more general context the following conjecture due to A. Zaharescu (2009): $\langle$For any $x,y\in\mathbf{C}_{p}$-the complex $p$-adic field, there exists $t\in\mathbf{Q}_{p}$-the $p$-adic number field, such that $\widetilde{\mathbf{Q}_{p}(x,y)}=\widetilde{\mathbf{Q}_{p}(x+ty)}$, where $\widetilde{L}$ means the $p$-adic topological closure of a subfield $L$ of $\mathbf{C}_{p}$ in $\mathbf{C}_{p}\rangle$.