Abstract
We give a description of the arboreal Galois representation of a quadratic polynomial \(f(x)=x^2-(4k+1)\), where \(k\in {\mathbb {N}}\) and \(4k+1\) is not a square in \({\mathbb {Z}}\).
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References
Cremona, J.E.: On the Galois groups of the iterates of \(x^2+1\). Mathematika 36, 259–261 (1989)
Gratton, C., Nguyen, K., Tucker, T.J.: ABC implies primitive prime divisors in arithmetic dynamic. Bull. Lond. Math. Soc. 45(6), 1194–1208 (2013)
Jones, R.: The density of prime divisors in the arithmetic dynamics of quadratic polynomials. J. Lond. Math. Soc. 78(2), 523–544 (2008)
Odoni, R.W.K.: The Galois theory of iterates and composites of polynomials. Proc. Lond. Math. Soc. 51(3), 385–414 (1985)
Odoni, R.W.K.: Realising wreath products of cyclic groups as Galois groups. Mathematika 35, 101–113 (1988)
Stoll, M.: Galois groups over \({\mathbb{Q}}\) of some iterated polynomials. Arch. Math. 59, 239–244 (1992)
Funding
Funding was provided by Ministry of Science and Technology, Taiwan (Grant No. 108-2115-M-003-001).
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Li, HC. Arboreal Galois representation for a certain type of quadratic polynomials. Arch. Math. 114, 265–269 (2020). https://doi.org/10.1007/s00013-019-01390-x
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DOI: https://doi.org/10.1007/s00013-019-01390-x