Almost hilbertian fields

Pierre Dèbes; Dan Haran

Acta Arithmetica (1999)

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

Abstract

top
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.

How to cite

top

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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.