Skip to main content
SpringerLink
Log in

Algebraic extensions of finite corank of hilbertian fields

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We consider here a hilbertian fieldk and its Galois group

(k s/k). For a natural numbere we prove that almost all (σ) ∈

(ks/k)e have the following properties. (1) The closedsubgroup 〈σ〉 which is generated by σ1, …, σe is a free pro-finite group withe generators. (2) LetK be a proper subfield of the fixed fieldk s (σ) of 〈σ〉, …, σe ink s, which containsk. Then the group

(k s/K) cannot be topologically generated by less thene+1 elements. (3) There does not exist a τ ∈

(k/k), τ≠1, of finite order such that [k s (σ):k s (σ, τ)]<∞. (4) Ife=1, there does not exist a fieldk⊆K⊆k s (σ) such that 1<[k s (σ):K]<∞. Here “almost all” is used in the sense of the Haar measure of the compact group

(ks/k)e.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Ax,Solving diophantic problems modulo every prime, Ann. of Math.85 (1967), 161–183.

    Article  MathSciNet  Google Scholar 

  2. J. Ax,The elementary theory of finite fields, Ann. of Math.88 (1968), 239–271.

    Article  MathSciNet  Google Scholar 

  3. E. Binz, J. Neukirch, G. H. Wenzel,A subgroup theorem for free products of pro-finite groups, J. Algebra19 (1971), 104–109.

    Article  MATH  MathSciNet  Google Scholar 

  4. S. Eilenberg and N. Steenrod,Foundations of Algebraic Topology, Princeton University Press, Princeton, New Jersey.

  5. W. D. Geyer,Unendliche algebraische Zahlkörper, über denen jede Gleichung auflösbar von beschränkter Stufe ist, J. Number Theory1 (1969), 346–374.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Hall, Jr.,Subgroups of finite index in free groups, Canad. J. Math.I, (1949), 187–190.

    Google Scholar 

  7. M. Hall, Jr.,A topology for free groups and related groups, Ann. of Math. (2)52 (1950), 127–139.

    Article  MathSciNet  Google Scholar 

  8. M. Jarden,Elementary statements over large algebraic fields, Trans. Amer. Math. Soc.164 (1972), 67–91.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. G. Kurosh,The Theory of Groups, Vol. II, second English edition, Chelsea Publishing Company, New York.

  10. W. Kuyk,Generic approach to the Galois embedding and extension problem, J. Algebra9 (1968), 393–407.

    Article  MATH  MathSciNet  Google Scholar 

  11. W. Kuyk,Extensions de corps hilbertiens, J. Algebra14 (1970), 112–124.

    Article  MATH  MathSciNet  Google Scholar 

  12. S. Lang,Introduction to algebraic geometry, Intersience tracts in pure and applied Mathematics, No. 5, Intersience Publishers, Inc., New York.

  13. S. Lang,Diophantine geometry, Intersience tracts in pure and applied Mathematics, No. 11, Intersience Publishers, new York, London.

  14. S. Lang,Algebra, Addison-Wesley Publishing Company, Reading, Massachusetts.

  15. L. Ribes,Introduction to pro-finite groups and Galois cohomology, Queens papers in pure and applied Mathematics, No. 24, 1970.

  16. J. P. Serre,Sur la dimension cohomologique des groupes profinis, Topology,3 (1964–65), 413–420.

    Article  MathSciNet  Google Scholar 

  17. J. R. Stalling,On torsion-free groups with infinitely many ends, Ann. of Math.88 (1968), 312–334.

    Article  MathSciNet  Google Scholar 

  18. A. Weil,Foundations of Algebraic Geometry, Revised and enlarged edition (1962), A. M. S. Colloquium publications Vol. XXIX Providence, Rhode Island.

Download references

Author information

Author notes

  1. Moshe Jarden

    Present address: Department of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, Israel

Authors and Affiliations

  1. Department of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

    Moshe Jarden

  2. Mathematisches Institut, Heidelberg Universität, Heidelberg, West Germany

    Moshe Jarden

Authors
  1. Moshe Jarden
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jarden, M. Algebraic extensions of finite corank of hilbertian fields. Israel J. Math. 18, 279–307 (1974). https://doi.org/10.1007/BF02757283

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02757283

Keywords

Search

Navigation