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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
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