Transcendental extensions of a valuation domain of rank one

Abstract

Let $V$ be a valuation domain of rank one and quotient field $K$. Let $\overline{\hat{K}}$ be a fixed algebraic closure of the $v$-adic completion $\hat{K}$ of $K$ and let $\overline{\hat{V}}$ be the integral closure of $\hat{V}$ in $\overline{\hat{K}}$. We describe a relevant class of valuation domains $W$ of the field of rational functions $K(X)$ which lie over $V$, which are indexed by the elements $\alpha \in \overline{\hat{K}}\cup \{\infty \}$, namely, $W=W_{\alpha }=\{\phi \in K(X)\mid \phi (\alpha )\in \overline{\hat{V}}\}$. If $V$ is discrete and $\pi \in V$ is a uniformizer, then a valuation domain $W$ of $K(X)$ is of this form if and only if the residue field degree $[W/M:V/P]$ is finite and $\pi W=M^e$, for some $e\ge 1$, where $M$ is the maximal ideal of $W$. In general, for $\alpha ,\beta \in \overline{\hat{K}}$ we have $W_{\alpha }=W_{\beta }$ if and only if $\alpha$ and $\beta$ are conjugated over $\hat{K}$. Finally, we show that the set ${\mathcal{P}}^{\mathrm{irr}}$ of irreducible polynomials over $\hat{K}$ endowed with an ultrametric distance introduced by Krasner is homeomorphic to the space $\{W_{\alpha }\mid \alpha \in \overline{\hat{K}}\}$ endowed with the Zariski topology.

accepted for publication in the Proceedings of the AMS (2016); comments are welcome!

Cited by in corpus (7)

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• The Zariski-Riemann space of valuation domains associated to pseudo-convergent sequences
• Polynomial Dedekind domains with finite residue fields of prime characteristic
• Isolated points of the Zariski space
• Stacked Pseudo-Convergent Sequences and Polynomial Dedekind Domains
• Metrizability of spaces of valuation domains associated to pseudo-convergent sequences