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arXiv · 1905.02481

Extending valuations to the field of rational functions using pseudo-monotone sequences

Giulio Peruginelli·Dario Spirito
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Abstract

Let $V$ be a valuation domain with quotient field $K$. We show how to describe all extensions of $V$ to $K(X)$ when the $V$-adic completion $\widehat{K}$ is algebraically closed, generalizing a similar result obtained by Ostrowski in the case of one-dimensional valuation domains. This is accomplished by realizing such extensions by means of pseudo-monotone sequences, a generalization of pseudo-convergent sequences introduced by Chabert. We also show that the valuation rings associated to pseudo-convergent and pseudo-divergent sequences (two classes of pseudo-monotone sequences) roughly correspond, respectively, to the closed and the open balls of $K$ in the topology induced by $V$.

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BibTeXRIS

Giulio Peruginelli, Dario Spirito. 2019-05-07. Extending valuations to the field of rational functions using pseudo-monotone sequences. https://doi.org/10.1016/j.jalgebra.2021.07.004

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