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Transcendental extensions of a valuation domain of rank one


Author: Giulio Peruginelli
Journal: Proc. Amer. Math. Soc. 145 (2017), 4211-4226
MSC (2010): Primary 16W60; Secondary 13J10, 13B25, 13F20
DOI: https://doi.org/10.1090/proc/13574
Published electronically: April 27, 2017
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Abstract: Let $ V$ be a valuation domain of rank one, maximal ideal $ P$ and quotient field $ K$. Let $ \overline {\widehat {K}}$ be a fixed algebraic closure of the $ v$-adic completion $ \widehat {K}$ of $ K$ and let $ \overline {\widehat {V}}$ be the integral closure of $ \widehat {V}$ in $ \overline {\widehat {K}}$. We describe a relevant class of valuation domains $ W$ of the field of rational functions $ K(X)$ lying over $ V$, which are indexed by the elements $ \alpha \in \overline {\widehat {K}}\cup \{\infty \}$, namely, the valuation domains $ W=W_{\alpha }=\{\varphi \in K(X) \mid \varphi (\alpha )\in \overline {\widehat {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\geq 1$, where $ M$ is the maximal ideal of $ W$. In general, for $ \alpha ,\beta \in \overline {\widehat {K}}$ we have $ W_{\alpha }=W_{\beta }$ if and only if $ \alpha $ and $ \beta $ are conjugate over $ \widehat {K}$. Finally, we show that the set $ {\mathcal {P}^{{\rm irr}}}$ of irreducible polynomials over $ \widehat {K}$ endowed with an ultrametric distance introduced by Krasner is homeomorphic to the space $ \{W_{\alpha } \mid \alpha \in \overline {\widehat {K}}\}$ endowed with the Zariski topology.


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Additional Information

Giulio Peruginelli
Affiliation: Dipartimento di Matematica, Universita’ di Pisa, Largo Pontecorvo 5, 56127 Pisa PI Italy
Address at time of publication: Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121 Padova, Italy
Email: gperugin@math.unipd.it

DOI: https://doi.org/10.1090/proc/13574
Received by editor(s): May 11, 2016
Received by editor(s) in revised form: September 24, 2016, and November 5, 2016
Published electronically: April 27, 2017
Additional Notes: The author was supported by the grant “Assegni Senior” of the University of Padova and by the grant “Assegno di ricerca Ing. G. Schirillo” of the Istituto Nazionale di Alta Matematica.
Communicated by: Irena Peeva
Article copyright: © Copyright 2017 American Mathematical Society

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