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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
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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  • [1] V. Alexandru, N. Popescu, and A. Zaharescu, All valuations on $ K(X)$, J. Math. Kyoto Univ. 30 (1990), no. 2, 281-296. MR 1068792
  • [2] N. Bourbaki, Algèbre commutative, Hermann, Paris, 1961.
  • [3] Jean-Luc Chabert and Giulio Peruginelli, Polynomial overrings of $ \mathrm {Int}(\mathbb{Z})$, J. Commut. Algebra 8 (2016), no. 1, 1-28. MR 3482343,
  • [4] Robert Gilmer, Multiplicative ideal theory, Queen's Papers in Pure and Applied Mathematics, vol. 90, Queen's University, Kingston, ON, 1992. Corrected reprint of the 1972 edition. MR 1204267
  • [5] Robert Gilmer, William Heinzer, David Lantz, and William Smith, The ring of integer-valued polynomials of a Dedekind domain, Proc. Amer. Math. Soc. 108 (1990), no. 3, 673-681. MR 1009989,
  • [6] Irving Kaplansky, Maximal fields with valuations, Duke Math. J. 9 (1942), 303-321. MR 0006161
  • [7] Marc Krasner, Nombre des extensions d'un degré donné d'un corps $ {\mathfrak{p}}$-adique, Les Tendances Géom. en Algèbre et Théorie des Nombres, Editions du Centre National de la Recherche Scientifique, Paris, 1966, pp. 143-169 (French). MR 0225756
  • [8] Franz-Viktor Kuhlmann, Value groups, residue fields, and bad places of rational function fields, Trans. Amer. Math. Soc. 356 (2004), no. 11, 4559-4600. MR 2067134,
  • [9] K. Alan Loper and Francesca Tartarone, A classification of the integrally closed rings of polynomials containing $ \mathbb{Z}[X]$, J. Commut. Algebra 1 (2009), no. 1, 91-157. MR 2462383,
  • [10] K. Alan Loper and Nicholas J. Werner, Generalized rings of integer-valued polynomials, J. Number Theory 132 (2012), no. 11, 2481-2490. MR 2954985,
  • [11] Saunders MacLane, A construction for absolute values in polynomial rings, Trans. Amer. Math. Soc. 40 (1936), no. 3, 363-395. MR 1501879,
  • [12] Saunders Mac Lane, A construction for prime ideals as absolute values of an algebraic field, Duke Math. J. 2 (1936), no. 3, 492-510. MR 1545943,
  • [13] Julio Fernández, Jordi Guàrdia, Jesús Montes, and Enric Nart, Residual ideals of MacLane valuations, J. Algebra 427 (2015), 30-75. MR 3312294,
  • [14] Giulio Peruginelli, The ring of polynomials integral-valued over a finite set of integral elements, J. Commut. Algebra 8 (2016), no. 1, 113-141. MR 3482349,
  • [15] Paulo Ribenboim, The theory of classical valuations, Springer Monographs in Mathematics, Springer-Verlag, New York, 1999. MR 1677964
  • [16] Michel Vaquié, Extension d'une valuation, Trans. Amer. Math. Soc. 359 (2007), no. 7, 3439-3481 (electronic) (French, with English summary). MR 2299463,
  • [17] Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, Springer-Verlag, New York-Heidelberg, 1975. Reprint of the 1960 edition; Graduate Texts in Mathematics, Vol. 29. MR 0389876

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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

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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