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On valuations of K(x)

Published online by Cambridge University Press:  20 January 2009

Sudesh K. Khanduja
Sudesh K. Khanduja
Affiliation:
Department of MathematicsPanjab University, ChandigarhIndia—160014
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Abstract

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For a valued field (K, v), let Kv denote the residue field of v and Gv its value group. One way of extending a valuation v defined on a field K to a simple transcendental extension K(x) is to choose any α in K and any μ in a totally ordered Abelian group containing Gv, and define a valuation w on K[x] by w(Σici(x – α)i) = mini (v(ci) + iμ). Clearly either Gv is a subgroup of finite index in Gw = Gv + ℤμ or Gw/Gv is not a torsion group. It can be easily shown that K(x)w is a simple transcendental extension of Kv in the former case. Conversely it is well known that for an algebraically closed field K with a valuation v, if w is an extension of v to K(x) such that either K(x)w is not algebraic over Kv or Gw/Gv is not a torsion group, then w is of the type described above. The present paper deals with the converse problem for any field K. It determines explicitly all such valuations w together with their residue fields and value groups.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 35 , Issue 3 , October 1992 , pp. 419 - 426
Copyright
Copyright © Edinburgh Mathematical Society 1992

References

REFERENCES

1.Bourbaki, N., Commutative Algebra, Chapter VI (Hermann, 1972).Google Scholar
2.Khanduja, S. K. and Garg, U., On extensions of valuations to simple transcendental extensions, Proc. Edinburgh Math. Soc. 32 (1989), 147–156.CrossRefGoogle Scholar
3.Nagata, M., A theorem on valuation rings and its applications, Nagoya Math. J. 29 (1967), 85–91.CrossRefGoogle Scholar
4.Ohm, J., Simple transcendental extensions of valued fields, J. Math. Kyoto Univ. 22 (1982), 201–221.Google Scholar
5.Ohm, J., The ruled residue theorem for simple transcendental extensions of valued fields, Proc. Amer. Math. Soc. 89 (1983), 16–18.CrossRefGoogle Scholar