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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Value groups, residue fields, and bad places of rational function fields


Author: Franz-Viktor Kuhlmann
Journal: Trans. Amer. Math. Soc. 356 (2004), 4559-4600
MSC (2000): Primary 12J10; Secondary 12J15, 16W60
Published electronically: May 28, 2004
MathSciNet review: 2067134
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03463-4
PII: S 0002-9947(04)03463-4
Received by editor(s): July 12, 2002
Received by editor(s) in revised form: July 15, 2003
Published electronically: May 28, 2004
Article copyright: © Copyright 2004 American Mathematical Society