Value groups, residue fields, and bad places of rational function fields
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Abstract:
We classify all possible extensions of a valuation from a ground field $K$ to a rational function field in one or several variables over $K$. We determine which value groups and residue fields can appear, and we show how to construct extensions having these value groups and residue fields. In particular, we give several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated. In the case of a rational function field $K(x)$ in one variable, we consider the relative algebraic closure of $K$ in the henselization of $K(x)$ with respect to the given extension, and we show that this can be any countably generated separable-algebraic extension of $K$. In the “tame case”, we show how to determine this relative algebraic closure. Finally, we apply our methods to power series fields and the $p$-adics.References
- Victor Alexandru and Nicolae Popescu, Sur une classe de prolongements à $K(X)$ d’une valuation sur un corps $K$, Rev. Roumaine Math. Pures Appl. 33 (1988), no. 5, 393–400 (French). MR 950136
- Victor Alexandru, Nicolae Popescu, and Alexandru Zaharescu, A theorem of characterization of residual transcendental extensions of a valuation, J. Math. Kyoto Univ. 28 (1988), no. 4, 579–592. MR 981094, DOI 10.1215/kjm/1250520346
- V. Alexandru, N. Popescu, and Al. Zaharescu, Minimal pairs of definition of a residual transcendental extension of a valuation, J. Math. Kyoto Univ. 30 (1990), no. 2, 207–225. MR 1068787, DOI 10.1215/kjm/1250520067
- V. Alexandru, N. Popescu, and A. Zaharescu, All valuations on $K(X)$, J. Math. Kyoto Univ. 30 (1990), no. 2, 281–296. MR 1068792, DOI 10.1215/kjm/1250520072
- Bourbaki, N.$\,$: Commutative algebra, Paris (1972)
- Steven Dale Cutkosky, Local monomialization and factorization of morphisms, Astérisque 260 (1999), vi+143 (English, with English and French summaries). MR 1734239
- Cutkosky, D. – Piltant, O.$\,$: Ramification of valuations, to appear in: Advances in Mathematics
- Otto Endler, Valuation theory, Universitext, Springer-Verlag, New York-Heidelberg, 1972. To the memory of Wolfgang Krull (26 August 1899–12 April 1971). MR 0357379
- B. Green, M. Matignon, and F. Pop, On valued function fields. I, Manuscripta Math. 65 (1989), no. 3, 357–376. MR 1015661, DOI 10.1007/BF01303043
- Green, B. – Matignon, M. – Pop, F.$\,$: On valued function fields II, III, J. reine angew. Math. 412 (1990), 128–149; 432 (1992), 117–133
- Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
- Sudesh K. Khanduja, Value groups and simple transcendental extensions, Mathematika 38 (1991), no. 2, 381–385 (1992). MR 1147837, DOI 10.1112/S0025579300006720
- Sudesh K. Khanduja, Prolongations of valuations to simple transcendental extensions with given residue field and value group, Mathematika 38 (1991), no. 2, 386–390 (1992). MR 1147838, DOI 10.1112/S0025579300006732
- Sudesh K. Khanduja, On valuations of $K(x)$, Proc. Edinburgh Math. Soc. (2) 35 (1992), no. 3, 419–426. MR 1187004, DOI 10.1017/S0013091500005708
- Sudesh K. Khanduja, A uniqueness problem in simple transcendental extensions of valued fields, Proc. Edinburgh Math. Soc. (2) 37 (1994), no. 1, 13–23. MR 1258026, DOI 10.1017/S0013091500018642
- Sudesh K. Khanduja, On value groups and residue fields of some valued function fields, Proc. Edinburgh Math. Soc. (2) 37 (1994), no. 3, 445–454. MR 1297313, DOI 10.1017/S0013091500018897
- Sudesh K. Khanduja, A note on residually transcendental prolongations with uniqueness property, J. Math. Kyoto Univ. 36 (1996), no. 3, 553–556. MR 1417826, DOI 10.1215/kjm/1250518509
- Sudesh K. Khanduja, On residually transcendental valued function fields of conics, Glasgow Math. J. 38 (1996), no. 2, 137–145. MR 1397168, DOI 10.1017/S0017089500031360
- Sudesh K. Khanduja, On extensions of valuations with prescribed value groups and residue fields, J. Indian Math. Soc. (N.S.) 62 (1996), no. 1-4, 57–60. MR 1458473
- Sudesh K. Khanduja, Valued function fields with given genus and residue field, J. Indian Math. Soc. (N.S.) 63 (1997), no. 1-4, 115–121. MR 1618074
- Sudesh K. Khanduja, An independence theorem in simple transcendental extensions of valued fields, J. Indian Math. Soc. (N.S.) 63 (1997), no. 1-4, 243–248. MR 1618022
- Sudesh K. Khanduja, Tame fields and tame extensions, J. Algebra 201 (1998), no. 2, 647–655. MR 1612351, DOI 10.1006/jabr.1997.7298
- Franz-Viktor Kuhlmann, Salma Kuhlmann, and Murray Marshall (eds.), Valuation theory and its applications. Vol. I, Fields Institute Communications, vol. 32, American Mathematical Society, Providence, RI, 2002. MR 1928358, DOI 10.1090/fic/032
- Sudesh K. Khanduja and Usha Garg, On extensions of valuations to simple transcendental extensions, Proc. Edinburgh Math. Soc. (2) 32 (1989), no. 1, 147–156. MR 982002, DOI 10.1017/S0013091500007008
- Sudesh K. Khanduja and Usha Garg, Rank $2$ valuations of $K(x)$, Mathematika 37 (1990), no. 1, 97–105. MR 1067891, DOI 10.1112/S0025579300012833
- Sudesh K. Khanduja and Usha Garg, On rank of extensions of valuations, Colloq. Math. 59 (1990), no. 1, 25–29. MR 1078288, DOI 10.4064/cm-59-1-25-29
- Sudesh K. Khanduja and Usha Garg, On residually generic prolongations of a valuation to a simple transcendental extension, J. Indian Math. Soc. (N.S.) 57 (1991), no. 1-4, 101–108. MR 1161327
- Sudesh K. Khanduja and Usha Garg, Residue fields of valued function fields of conics, Proc. Edinburgh Math. Soc. (2) 36 (1993), no. 3, 469–478. MR 1242758, DOI 10.1017/S0013091500018551
- Sudesh K. Khanduja and Usha Garg, Prolongations of a Krull valuation to a simple transcendental extension, J. Indian Math. Soc. (N.S.) 59 (1993), no. 1-4, 13–21. MR 1248942
- Khanduja, S. K. – Popescu, N. – Roggenkamp, K. W.$\,$: On minimal pairs and residually transcendental extensions of valuations, to appear in: Mathematika
- Sudesh K. Khanduja and Jayanti Saha, A uniqueness problem in valued function fields of conics, Bull. London Math. Soc. 28 (1996), no. 5, 455–462. MR 1396143, DOI 10.1112/blms/28.5.455
- Knaf, H. – Kuhlmann, F.–V.$\,$: Abhyankar places admit local uniformization in any characteristic, submitted, available at: http://math.usask.ca/fvk/Valth.html
- Knaf, H. – Kuhlmann, F.–V.$\,$: Every place admits local uniformization in a finite extension of the function field, preprint available at: http://math.usask.ca/fvk/Valth.html
- Kuhlmann, F.–V.$\,$: Henselian function fields and tame fields, preprint (extended version of Ph.D. thesis), Heidelberg (1990)
- Kuhlmann, F.–V.$\,$: Valuation theory of fields, abelian groups and modules, to appear in the “Algebra, Logic and Applications” series (formerly Gordon and Breach, eds. A. Macintyre and R. Göbel). Preliminary versions of several chapters available at: http://math.usask.ca/~fvk/Fvkbook.htm
- Franz-Viktor Kuhlmann, Valuation theoretic and model theoretic aspects of local uniformization, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 381–456. MR 1748629
- Kuhlmann, F.–V.$\,$: The model theory of tame valued fields, in preparation
- Kuhlmann, F.–V.$\,$: Algebraic independence of elements in completions and maximal immediate extensions of valued fields, in preparation
- F.-V. Kuhlmann, S. Kuhlmann, M. Marshall, and M. Zekavat, Embedding ordered fields in formal power series fields, J. Pure Appl. Algebra 169 (2002), no. 1, 71–90. MR 1890186, DOI 10.1016/S0022-4049(01)00064-0
- Franz-Viktor Kuhlmann, Matthias Pank, and Peter Roquette, Immediate and purely wild extensions of valued fields, Manuscripta Math. 55 (1986), no. 1, 39–67. MR 828410, DOI 10.1007/BF01168612
- Manfred Knebusch and Michael J. Wright, Bewertungen mit reeller Henselisierung, J. Reine Angew. Math. 286(287) (1976), 314–321 (German). MR 419419
- T. Y. Lam, The theory of ordered fields, Ring theory and algebra, III (Proc. Third Conf., Univ. Oklahoma, Norman, Okla., 1979) Lecture Notes in Pure and Appl. Math., vol. 55, Dekker, New York, 1980, pp. 1–152. MR 584611
- MacLane, S.$\,$: A construction for absolute values in polynomial rings, Trans. Amer. Math. Soc. 40 (1936), 363–395
- Michel Matignon and Jack Ohm, A structure theorem for simple transcendental extensions of valued fields, Proc. Amer. Math. Soc. 104 (1988), no. 2, 392–402. MR 962804, DOI 10.1090/S0002-9939-1988-0962804-0
- Michel Matignon and Jack Ohm, Simple transcendental extensions of valued fields. III. The uniqueness property, J. Math. Kyoto Univ. 30 (1990), no. 2, 347–365. MR 1068796, DOI 10.1215/kjm/1250520076
- T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
- Mosteig, E. – Sweedler, M.$\,$: Well-ordered valuations on characteristic zero rational function fields of transcendence degree two, manuscript (2001), available at: http://math.usask.ca/fvk/Valth.html
- Edward Mosteig and Moss Sweedler, Valuations and filtrations, J. Symbolic Comput. 34 (2002), no. 5, 399–435. MR 1937467, DOI 10.1006/jsco.2002.0565
- Masayoshi Nagata, A theorem on valuation rings and its applications, Nagoya Math. J. 29 (1967), 85–91. MR 207688
- Jack Ohm, Simple transcendental extensions of valued fields, J. Math. Kyoto Univ. 22 (1982/83), no. 2, 201–221. MR 666971, DOI 10.1215/kjm/1250521810
- Jack Ohm, The ruled residue theorem for simple transcendental extensions of valued fields, Proc. Amer. Math. Soc. 89 (1983), no. 1, 16–18. MR 706500, DOI 10.1090/S0002-9939-1983-0706500-9
- Jack Ohm, Simple transcendental extensions of valued fields. II. A fundamental inequality, J. Math. Kyoto Univ. 25 (1985), no. 3, 583–596. MR 807499, DOI 10.1215/kjm/1250521073
- Marc Polzin, Prolongement de la valeur absolue de Gauss et problème de Skolem, Bull. Soc. Math. France 116 (1988), no. 1, 103–132 (French, with English summary). MR 946280
- Liliana Popescu and Nicolae Popescu, On the residual transcendental extensions of a valuation. Key polynomials and augmented valuation, Tsukuba J. Math. 15 (1991), no. 1, 57–78. MR 1118582, DOI 10.21099/tkbjm/1496161567
- Alexander Prestel, Lectures on formally real fields, Lecture Notes in Mathematics, vol. 1093, Springer-Verlag, Berlin, 1984. MR 769847, DOI 10.1007/BFb0101548
- Paulo Ribenboim, Théorie des valuations, Séminaire de Mathématiques Supérieures, No. 9 (Été, vol. 1964, Les Presses de l’Université de Montréal, Montreal, Que., 1968 (French). Deuxième édition multigraphiée. MR 0249425
- Mark Spivakovsky, Valuations in function fields of surfaces, Amer. J. Math. 112 (1990), no. 1, 107–156. MR 1037606, DOI 10.2307/2374856
- Sweedler, M.$\,$: Ideal Bases and Valuation Rings, manuscript (1986), available at: http://myweb.lmu.edu/faculty/emosteig/
- Vaquié, M.$\,$: Extension d’une valuation, preprint (2002), available at: http://math.usask.ca/fvk/Valth.html
- Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0120249
Additional Information
- Franz-Viktor Kuhlmann
- Affiliation: Mathematical Sciences Group, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada S7N 5E6
- Email: fvk@math.usask.ca
- Received by editor(s): July 12, 2002
- Received by editor(s) in revised form: July 15, 2003
- Published electronically: May 28, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 4559-4600
- MSC (2000): Primary 12J10; Secondary 12J15, 16W60
- DOI: https://doi.org/10.1090/S0002-9947-04-03463-4
- MathSciNet review: 2067134