Mathematika

Research Article

Rank 2 valuations of K(x)

Sudesh K. Khandujaa1 and Usha Garga2

a1 Centre for Advanced Study in Mathematics, Panjab University, Chandigarh—160014, India.

a2 Centre for Advanced Study in Mathematics, Panjab University, Chandigarh—160014, India.

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.

(Received July 17 1989)