Vol. 70, No. 1, 1977

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Locally bounded topologies on F(X)

Jo-Ann Deborah Cohen

Vol. 70 (1977), No. 1, 125–132

It is classic that, to within equivalence, the only valuations on the field F(X) of rational functions over a field F that are improper on F are the valuations vp, where p is a prime of the principal ideal subdomain F[X] of F(X), and the valuation v, defined by the prime X1 of the principal ideal subdomain F[X1] of F(X) ([1], p. 94, Corollary 2). If 𝒯 is the supremum of finitely many of the associated valuation topologies, then 𝒯 is a Hausdorff, locally bounded ring topology on F(X) for which F is a bounded set and for which there is a nonzero topological nilpotent a. In this paper we shall show conversely that any topology on F(X) having these properties is the supremum of finitely many valuation topologies.

Mathematical Subject Classification 2000
Primary: 12J99
Received: 29 October 1976
Revised: 7 January 1977
Published: 1 May 1977
Jo-Ann Deborah Cohen