The strong approximation theorem and locally bounded topologies on F(X)

Jo-Ann Deborah Cohen

Vol. 87 (1980), No. 1, 59–63

DOI: 10.2140/pjm.1980.87.59

Abstract

To within equivalence, the only valuations on the field F(X) of rational functions over F that are improper on F are the valuations v_p, where p is a prime polynomial of F[X], and the valuation v_∞, defined by the prime polynomial X⁻¹ of F[X⁻¹]. It is classic that if F is a finite field, the set 𝒫′ defined by, 𝒫′ = {p : p is a prime polynomial over Fi}∪{∞}, has the Strong Approximation Property, that is, for any finite subset G of 𝒫′, any q ∈𝒫′∖ G, any family (a_g)_g∈G of elements of F(X) indexed by G, and any M > 0, there exists a nonzero element h in F(X) such that v_p(h−a_p) > M for all p in G and v_p(h) ≧ 0 for all p in 𝒫′∖ (G ∪{q}). We shall first prove that 𝒫′ satisfies this condition when F is infinite as well. We then apply this result to obtain a characterization of all locally bounded topologies on F(X) for which the subfield F is bounded.

Mathematical Subject Classification 2000

Primary: 12J99

Secondary: 13J99

Milestones

Received: 15 May 1978

Published: 1 March 1980