On cuts of the quotient field of a ring of formal power series

N. Yu. Galanova

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 83 (2023), pp. 5-16

Résumé

In studies related to the classification of real-closed fields, fields of formal power series with a multiplicative divisible group of Archimedean classes are essentially used. Consider a linearly ordered Abelian divisible group $G = G(L,\mathbf{Q})$, which consists of words with generators from a linearly ordered set $L$ similar to the ordinal $\omega_1$ and rational exponents. The article deals with the properties of sections of subfields of the field of bounded formal power series $\mathbf{R}[[G,\aleph_1]]$. For all $\xi_i\in L$ we set $t_i=\xi_i^{-1}$. Consider an infinite strictly decreasing sequence $\{t_\gamma\}_{\gamma\in\Gamma}$, where $\Gamma\subseteq\omega_1\setminus\{1\}$ is an arbitrary infinite set. Series of the form $x = \sum\limits_{\gamma\in\Gamma} r_\gamma\cdot t_\gamma\in \mathbf{R}[[G]]$, where $r_\gamma\ne0$ for all $\gamma\in\Gamma$, i.e. $\mathrm{supp}(x) = \{t_\gamma \mid \gamma\in\Gamma\}$, we will call series of the form ($*$). We prove that series of the form ($*$) for $r_\gamma$ for all $\gamma\in\Gamma$ generate in the field $qf\mathbf{R}[[G,\aleph_0]] = K$ symmetric non-fundamental sections of confinality $(\aleph_0,\aleph_0)$, in the real closure $\overline{qf\mathbf{R}[[G,\aleph_0]]}= \overline{K}$ series ($*$) generate symmetric sections. Let $H$ be the least by inclusion real closed subfield of the field $\mathbf{R}[[G,\aleph_1]]$ containing $\overline{K}$ and all truncations of the series $x_{\omega_1}=\sum\limits_{\gamma\in\omega_1}1\cdot t_\gamma$. Then $\overline{K}\ne H$ and the elements of the real closure of the simple transcendental extension $\overline{H(x_{\omega_1})}$ that do not belong to $H$ generate symmetric sections of the type $(\aleph_1,\aleph_1)$ in the field $H$.

Détail

Keywords: divisible totally ordered Abelian group, real closed field, field of bounded formal (generalized) power series, symmetric cut (non-ball cut), cofinality of a cut, fundamental cut (Scott cut), quotient field.

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