Skip to main content
SpringerLink
Log in

Computable valued fields

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

We investigate the computability-theoretic properties of valued fields, and in particular algebraically closed valued fields and p-adically closed valued fields. We give an effectiveness condition, related to Hensel’s lemma, on a valued field which is necessary and sufficient to extend the valuation to any algebraic extension. We show that there is a computable formally p-adic field which does not embed into any computable p-adic closure, but we give an effectiveness condition on the divisibility relation in the value group which is sufficient to find such an embedding. By checking that algebraically closed valued fields and p-adically closed valued fields of infinite transcendence degree have the Mal’cev property, we show that they have computable dimension \(\omega \).

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. Calvert, W.: The isomorphism problem for classes of computable fields. Arch. Math. Logic 43(3), 327–336 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chatzidakis, Z.: Introductory notes on the model theory of valued fields. In: Cluckers, R., Nicaise, J., Sebag, J. (eds.) Motivic Integration and Its Interactions with Model Theory and Non-Archimedean Geometry, vol I, volume 383 of London Mathematical Society Lecture Note Series, pp. 35–79. Cambridge University Press, Cambridge (2011)

  3. Denef, J.: \(p\)-adic semi-algebraic sets and cell decomposition. J. Reine Angew. Math. 369, 154–166 (1986)

    MathSciNet  MATH  Google Scholar 

  4. Dobritsa, V.P.: Some constructivizations of abelian groups. Sibirsk. Mat. Zh. 24(2), 18–25 (1983)

    MathSciNet  MATH  Google Scholar 

  5. Engler, A.J., Prestel, A.: Valued Fields. Springer Monographs in Mathematics. Springer, Berlin (2005)

    MATH  Google Scholar 

  6. Goncharov, S.S., Lempp, S., Solomon, R.: The computable dimension of ordered abelian groups. Adv. Math. 175(1), 102–143 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Goncharov, S.S.: Limit equivalent constructivizations. In: Mathematical Logic and the Theory of Algorithms, volume 2 of Trudy Institute of Mathematics, pp. 4–12. “Nauka” Sibirsk. Otdel., Novosibirsk (1982)

  8. Harizanov, V.S.: Pure computable model theory. In: Ershov, Y.L., Goncharov, S.S., Nerode, A., Remmel, J.B. (eds) Handbook of recursive mathematics, vol. 1, volume 138 of Studies in Logic and the Foundations of Mathematics, pp. 3–114. North-Holland, Amsterdam (1998)

  9. Harrison-Trainor, M., Melnikov, A., Miller, R.: On computable field embeddings and difference closed fields. Can. J. Math. (to appear)

  10. Harrison-Trainor, M., Melnikov, A., Montalbán, A.: Independence in computable algebra. J. Algebra 443, 441–468 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Harrison-Trainor, M., Melnikov, A., Miller, R., Montalbán, A.: Computable functors and effective interpretability. J. Symb. Logic 82(1), 77–97

  12. Igusa, G., Knight, J., Schweber, N.: Computing strength of structures related to the field of real numbers. J. Symb. Logic 82(1), 137–150

  13. Knight, J.F., Lange, K.: Complexity of structures associated with real closed fields. Proc. Lond. Math. Soc. (3) 107(1), 177–197 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kronecker, L.: Grundzüge einer arithmetischen theorie der algebraischen größen. J. f. Math 92, 1–122 (1882)

    MathSciNet  MATH  Google Scholar 

  15. Macintyre, A.: On definable subsets of \(p\)-adic fields. J. Symb. Logic 41(3), 605–610 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  16. Miller, R.: Computable fields and Galois theory. Notices Am. Math. Soc. 55(7), 798–807 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Marker, D., Miller, R.: Turing degree spectra of differentially closed fields. J. Symb. Logic 82(1), 1–25

  18. Miller, R., Ovchinnikov, A., Trushin, D.: Computing constraint sets for differential fields. J. Algebra 407, 316–357 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mourgues, M.-H.: Cell decomposition for \(P\)-minimal fields. MLQ Math. Log. Q. 55(5), 487–492 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nurtazin, A.T.: Computable classes and algebraic criteria of autostability. Ph.D. thesis, Novosibirsk (1974)

  21. Ocasio, V.A.: Computability in the class of real closed fields. ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.), University of Notre Dame

  22. Prestel, A., Roquette, P.: Formally \(p\)-adic Fields, volume 1050 of Lecture Notes in Mathematics. Springer, Berlin (1984)

    MATH  Google Scholar 

  23. Rabin, M.O.: Computable algebra, general theory and theory of computable fields. Trans. Am. Math. Soc. 95, 341–360 (1960)

    MathSciNet  MATH  Google Scholar 

  24. Smith, R.L.: Effective valuation theory. In: Crossley, J. N. (ed.) Aspects of Effective Algebra (Clayton, 1979), pp. 232–245. Upside Down A Book Co., Yarra Glen (1981)

  25. Scowcroft, P., van den Dries, L.: On the structure of semialgebraic sets over \(p\)-adic fields. J. Symb. Logic 53(4), 1138–1164 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  26. van den Dries, L.: Algebraic theories with definable Skolem functions. J. Symb. Logic 49(2), 625–629 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  27. van der Waerden, B.L.: Algebra, vol 1. Translated by Fred Blum and John R. Schulenberger. Frederick Ungar Publishing Co., New York (1970)

Download references

Author information

Author notes

  1. Matthew Harrison-Trainor

    Present address: Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada

Authors and Affiliations

  1. Group in Logic and the Methodology of Science, University of California, Berkeley, CA, USA

    Matthew Harrison-Trainor

Authors
  1. Matthew Harrison-Trainor
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthew Harrison-Trainor.

Additional information

The author was partially supported by the Berkeley Fellowship and NSERC Grant PGSD3-454386-2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Harrison-Trainor, M. Computable valued fields. Arch. Math. Logic 57, 473–495 (2018). https://doi.org/10.1007/s00153-017-0589-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-017-0589-9

Keywords

Mathematics Subject Classification

Search

Navigation