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Rank 2 valuations of K(x)

  • Sudesh K. Khanduja (a1) and Usha Garg (a1)
Abstract

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.

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3. S. MacLane . A construction for absolute values in polynomial rings. Trans. Amer. Math. Soc, 40 (1936), 363395.

5. O. Zariski . The reduction of the singularities of an algebraic surface. Annals of Math., 40 (1939), 639689.

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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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