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On extensions of valuations to simple transcendental extensions

Published online by Cambridge University Press:  20 January 2009

Sudesh K. Khanduja  and
Usha Garg
Sudesh K. Khanduja
Affiliation:
Department of MathematicsPanjab UniversityChandigarh-160014India
Usha Garg
Affiliation:
Department of MathematicsPanjab UniversityChandigarh-160014India
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Let ν0 be a valuation of a field K0 with residue field k0 and value group Z, the group of rational integers. Let K0(x) be a simple transcendental extension of K0. In 1936, Maclane [3] gave a method to determine all real valuations V of K0(x) which are extensions of ν0. But his method does not seem to give an explicit construction of these valuations. In the present paper, assuming K0 to be a complete field with respect to ν0, we explicitly determine all extensions of ν0 to K0(x) which have Z as the value group and a simple transcendental extension of k0 as the residue field. If V is any extension of ν0 to K0(x) having Z as the value group and a transcendental extension of k0 as the residue field, then using the Ruled Residue theorem [4, 2, 5], we give a method which explicitly determines V on a subfield of K0(x) properly containing K0.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 32 , Issue 1 , February 1989 , pp. 147 - 156
Copyright
Copyright © Edinburgh Mathematical Society 1989

References

REFERENCES

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