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Arithmetic properties of periodic points of quadratic maps. (English) Zbl 0767.11016
Let $$f$$ be a polynomial over a field $$k$$, denote by $$f_ n$$ the $$n$$th iterate of $$f$$ and define $$\Phi_{d,f}\in k[X]$$ by $$f_ n(X)- X=\prod_{d\mid n} \Phi_{d,f}(X)$$. The author studies the factorization of $$F_ f=\Phi_{3,f}$$ in case of $$f$$ quadratic and obtains several results concerning the corresponding Galois group. In particular, in case $$f(X)=X^ 2+a$$ ($$a$$ rational, with $$-(4a+7)$$ not a rational square) the Galois group of $$F_ f$$ turns out to be either $$C_ 6$$ or $$S_ 3$$ or the full wreath product of cyclic groups of order 2 and 3.

MSC:
 37P35 Arithmetic properties of periodic points 11R09 Polynomials (irreducibility, etc.) 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps 11R32 Galois theory
Keywords:
polynomial iteration; Galois group
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