# Almost hilbertian fields

Acta Arithmetica (1999)

- Volume: 88, Issue: 3, page 269-287
- ISSN: 0065-1036

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topPierre Dèbes, and Dan Haran. "Almost hilbertian fields." Acta Arithmetica 88.3 (1999): 269-287. <http://eudml.org/doc/207246>.

@article{PierreDèbes1999,

abstract = {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.},

author = {Pierre Dèbes, Dan Haran},

journal = {Acta Arithmetica},

keywords = {Hilbert specialization property; inverse Galois problem},

language = {eng},

number = {3},

pages = {269-287},

title = {Almost hilbertian fields},

url = {http://eudml.org/doc/207246},

volume = {88},

year = {1999},

}

TY - JOUR

AU - Pierre Dèbes

AU - Dan Haran

TI - Almost hilbertian fields

JO - Acta Arithmetica

PY - 1999

VL - 88

IS - 3

SP - 269

EP - 287

AB - 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.

LA - eng

KW - Hilbert specialization property; inverse Galois problem

UR - http://eudml.org/doc/207246

ER -

## References

top- [DeDes] P. Dèbes and B. Deschamps, The regular Inverse Galois Problem over large fields, in: Geometric Galois Actions, London Math. Soc. Lecture Note Ser. 243, Cambridge Univ. Press, 1997, 119-138. Zbl0905.12004
- [Des] B. Deschamps, Corps pythagoriciens, corps P-réduisants, in: Proc. '1997 Journées arithmétiques' in Limoges, to appear.
- [FrJa] M. D. Fried and M. Jarden, Field Arithmetic, Ergeb. Math. Grenzgeb. (3) 11, Springer, Heidelberg, 1986.
- [FrVo] M. Fried and H. Völklein, The embedding problem over a Hilbertian PAC-field, Ann. of Math. 135 (1992), 469-481. Zbl0765.12002
- [Ha] D. Haran, Hilbertian fields under separable algebraic extensions, Invent. Math., to appear. Zbl0933.12003
- [HaJa] D. Haran and M. Jarden, Regular split embedding problems over complete valued fields, Forum Math. 10 (1998), 329-351. Zbl0903.12003
- [La] S. Lang, Algebra, 3rd ed., Addison-Wesley, Reading, 1994.
- [Mas] W. S. Massey, Algebraic Topology: An Introduction, Springer, New York, 1984.
- [Mat] B. H. Matzat, Konstruktive Galoistheorie, Lecture Notes in Math. 1284, Springer, Berlin, 1987.
- [Po] F. Pop, Embedding problems over large fields, Ann. of Math. 144 (1996), 1-35. Zbl0862.12003
- [Ri] P. Ribenboim, Pythagorean and fermatian fields, Symposia Math. 15 (1975), 469-481.

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