Vol. 87, No. 1, 1980

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

Jo-Ann Deborah Cohen

Vol. 87 (1980), No. 1, 59–63

To within equivalence, the only valuations on the field F(X) of rational functions over F that are improper on F are the valuations vp, where p is a prime polynomial of F[X], and the valuation v, defined by the prime polynomial X1 of F[X1]. 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 (ag)gG of elements of F(X) indexed by G, and any M > 0, there exists a nonzero element h in F(X) such that vp(hap) > M for all p in G and vp(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
Received: 15 May 1978
Published: 1 March 1980
Jo-Ann Deborah Cohen