Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-16T21:13:51.392Z Has data issue: false hasContentIssue false
  • English
  • Français

Rank 2 valuations of K(x)

Published online by Cambridge University Press:  26 February 2010

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.
Get access

Extract

Let Vo be a discrete real valuation of a field K and x an indeterminate. In 1936, MacLane [3] gave a method of constructing all real valuations of K(x) which are extensions of Vo. In this paper, we determine explicitly all rank 2 valuations of K(x) which extend Vo. One can thereby describe all rank 2 valuations of K(x, y) which are trivial on an arbitrary K; x, y being algebraically independent over the field K. The latter valuations have been considered by Zariski [5] in the case when K is an algebraically closed field of characteristic zero.

Type
Research Article
Information
Mathematika , Volume 37 , Issue 1 , June 1990 , pp. 97 - 105
Copyright
Copyright © University College London 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bourbaki, N.. Commutative Algebra (Hermann, 1972).Google Scholar
2.Iyanaga, S.. The Theory of Numbers (North-Holland, Oxford, 1975).Google Scholar
3.MacLane, S.. A construction for absolute values in polynomial rings. Trans. Amer. Math. Soc, 40 (1936), 363–395.CrossRefGoogle Scholar
4.Ohm, J.. Simple transcendental extensions of valued fields II: A fundamental inequality. J. Math. Kyoto Univ., 25 (1985), 583–596.Google Scholar
5.Zariski, O.. The reduction of the singularities of an algebraic surface. Annals of Math., 40 (1939), 639–689.CrossRefGoogle Scholar
6.Zariski, O. and Samuel, P.. Commutative Algebra II (Van Nostrand, East-West student edition, 1969).Google Scholar