Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-17T18:12:17.557Z Has data issue: false hasContentIssue false
  • English
  • Français

Residue fields of valued function fields of conics

Published online by Cambridge University Press:  20 January 2009

Sudesh K. Khanduja  and
Usha Garg
Sudesh K. Khanduja
Affiliation:
Centre for Advanced Study in Mathematics, Panjab University, Chandigarh—160014, India
Usha Garg
Affiliation:
Centre for Advanced Study in Mathematics, Panjab University, Chandigarh—160014, India
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Suppose that K is a function field of a conic over a subfield K0. Let v0 be a valuation of K0 with residue field k0 of characteristic ≠2. Let v be an extension of v0 to K having residue field k. It has been proved that either k is an algebraic extension of k0 or k is a regular function field of a conic over a finite extension of k0. This result can also be deduced from the genus inequality of Matignon (cf. [On valued function fields I, Manuscripta Math. 65 (1989), 357–376]) which has been proved using results about vector space defect and methods of rigid analytic geometry. The proof given here is more or less self-contained requiring only elementary valuation theory.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 36 , Issue 3 , October 1993 , pp. 469 - 478
Copyright
Copyright © Edinburgh Mathematical Society 1993

References

REFERENCES

1.Artin, E., Algebraic Numbers and Algebraic Functions (Gordon and Breach, New York, 1967).Google Scholar
2.Bourbaki, N., Commutative Algebra (Hermann, Publishers in Arts and Science, 1972).Google Scholar
3.Endler, O., Valuation Theory (Springer-Verlag, New York, 1972).CrossRefGoogle Scholar
4.Green, B., Matignon, M. and Pop, F., On valued function fields I, Manuscripta Math. 65 (1989), 357–376.Google Scholar
5.Handuja, S. K. K., On valuations of K(x), Proc. Edinburgh Math. Soc. 35 (1992), 419–426.CrossRefGoogle Scholar
6.Matignon, M., Genre et genre residuel des corps de fonctions values, Manuscripta Math. 58 (1987), 179–214.Google Scholar
7.Matignon, M. and Ohm, J., A structure theorem for simple transcendental extensions of valued fields, Proc. Amer. Math. Soc. 104 (1988), 392–402.Google Scholar
8.Ohm, J., The ruled residue theorem for simple transcendental extensions of valued fields, Proc. Amer. Math. Soc. 89 (1983), 16–18.Google Scholar
9.Ohm, J., Simple transcendental extensions of valued fields, J. Math. Kyoto Univ. 22 (1982), 201–221.Google Scholar
10.Ohm, J., Simple transcendental extensions of valued fields II: A fundamental inequality, J. Math. Kyoto Univ. 25 (1985), 583–596.Google Scholar
11.Ohm, J., Function Fields of Conies, A Theorem of Amitsur-MacRae, and a Problem of Zariski (Proc. of the Abhyankar 60th birthday conference, 1990, Springer-Verlag), to appear.Google Scholar
12.Weil, A., Foundations of Algebraic Geometry (Amer. Math. Soc. Colloq. Pub. 29, Providence, 1962).Google Scholar
13.Zariski, O. and Samuel, P., Commutative Algebra, 1 (van Nostrand, Princeton, 1955).Google Scholar