Skip to main content
SpringerLink
Log in

On Stoll’s criterion for the maximality of quadratic arboreal Galois representations

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

We show that for a quadratic polynomial \(f(x)=x^2-c\), where \(c=(8k+2)(8k+3)\) or \(c=(4k+1)(4k+2)+1\) with \(k\in {\mathbb {N}}\cup \{0\}\), the Galois group of the splitting field of each iterate \(f^n\) of f is isomorphic to the automorphism group of a complete binary rooted tree of height n.

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. Cremona, J.E.: On the Galois groups of the iterates of \(x^2+1\). Mathematika 36, 259–261 (1989)

    Article  MathSciNet  Google Scholar 

  2. Gratton, C., Nguyen, K., Tucker, T.J.: ABC implies primitive prime divisors in arithmetic dynamic. Bull. Lond. Math. Soc. 6(45), 1194–1208 (2013)

    Article  MathSciNet  Google Scholar 

  3. Hindes, W.: Points on elliptic curves parametrizing dynamical Galois groups. Acta Arith. 159, 149–167 (2013)

    Article  MathSciNet  Google Scholar 

  4. Jones, R.: The density of prime divisors in the arithmetic dynamics of quadratic polynomials. J. Lond. Math. Soc. 78, 523–544 (2008)

    Article  MathSciNet  Google Scholar 

  5. Jones, R.: Galois representations from pre-image trees: an arboreal survey. In: Actes de la Conférence “Théorie des Nombres et Applications”, pp. 107–136. Publ. Math. Besançon Alg\(\grave{\rm{e}}\)bre Théorie Nr., 2013. Presses Univ. Franche-Comté, Besançon (2013)

  6. Li, H.-C.: Arboreal Galois representation for a certain type of quadratic polynomials. Arch. Math. (Basel) 114, 265–269 (2020)

    Article  MathSciNet  Google Scholar 

  7. Odoni, R.W.K.: Realising wreath products of cyclic groups as Galois groups. Mathematika 35, 101–113 (1988)

    Article  MathSciNet  Google Scholar 

  8. Stoll, M.: Galois groups over \({{\mathbb{Q}}}\) of some iterated polynomials. Arch. Math. (Basel) 59, 239–244 (1992)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, National Taiwan Normal University, Taipei, ROC, Taiwan

    Hua-Chieh Li

Authors
  1. Hua-Chieh Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hua-Chieh Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, HC. On Stoll’s criterion for the maximality of quadratic arboreal Galois representations. Arch. Math. 117, 133–140 (2021). https://doi.org/10.1007/s00013-021-01609-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-021-01609-w

Keywords

Mathematics Subject Classification

Search

Navigation