Almost hilbertian fields

• Autorzy: Pierre Dèbes , Dan Haran
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 12F10: Separable extensions, Galois theory
• 12F12: Inverse Galois theory
• 12E25: Hilbertian fields; Hilbert's irreducibility theorem

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1999
• Tom: 88
• Numer: 3
• Strony: 269-287

Daty

wydano

1999

otrzymano

1998-07-07

Twórcy

autor

Pierre Dèbes

• Université des Sciences et Technologies de Lille, Mathématiques, 59655 Villeneuve d'Ascq Cedex, France

autor

Dan Haran

• School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel

Bibliografia

• [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.
• [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.
• [Ha] D. Haran, Hilbertian fields under separable algebraic extensions, Invent. Math., to appear.
• [HaJa] D. Haran and M. Jarden, Regular split embedding problems over complete valued fields, Forum Math. 10 (1998), 329-351.
• [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.
• [Ri] P. Ribenboim, Pythagorean and fermatian fields, Symposia Math. 15 (1975), 469-481.