Geodesic
Almost hilbertian fields
Acta Arithmetica, Tome 88 (1999) no. 3, pp. 269-287 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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This paper is devoted to some variants of the Hilbert specialization property. For example, the RG-hilbertian property (for a field K), which arose in connection with the Inverse Galois Problem, requires that the specialization property holds solely for extensions of K(T) that are Galois and regular over K. We show that fields inductively obtained from a real hilbertian field by adjoining real pth roots (p odd prime) are RG-hilbertian; some of these fields are not hilbertian. There are other variants of interest: the R-hilbertian property is obtained from the RG-hilbertian property by dropping the condition "Galois", the mordellian property is that every non-trivial extension of K(T) has infinitely many non-trivial specializations, etc. We investigate the connections existing between these properties. In the case of PAC fields we obtain pure Galois-theoretic characterizations. We use them to show that "mordellian" does not imply "hilbertian" and that every PAC R-hilbertian field is hilbertian.
DOI : 10.4064/aa-88-3-269-287

Pierre Dèbes 1 ; Dan Haran 1

1 
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Pierre Dèbes; Dan Haran. Almost hilbertian fields. Acta Arithmetica, Tome 88 (1999) no. 3, pp. 269-287. doi: 10.4064/aa-88-3-269-287

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