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On the Galois groups of the iterates of \(x^ 2+1\). (English) Zbl 0699.12018
Let \(f_ 1(x)=x^ 2+1\) and \(f_ n(x)=f_ 1(f_{n-1}(x))\) for \(n\geq 2\). Let \(K_ n\) be the splitting field of \(f_ n(x)\) over \(\mathbb Q\) and \(\Omega_ n=\text{Gal}(K_ n/\mathbb Q)\). R. W. K. Odoni [Mathematika 35, No. 1, 101–113 (1988; Zbl 0662.12010)] proved that \(\Omega_ n\) is a subgroup of the \(n\)-th wreath power of \(\mathbb Z/2\mathbb Z\) and gave a simple rational criterion for \(\Omega_ n\) to be isomorphic to the \(n\)-th wreath power of \(\mathbb Z/2\mathbb Z\). The author describes a computer program for Odoni’s criterion and states that for all \(n\leq 5\cdot 10^ 7\), \(\Omega_ n\) is isomorphic to the \(n\)-th wreath power of \(\mathbb Z/2\mathbb Z\).

MSC:
11R32 Galois theory
11-04 Software, source code, etc. for problems pertaining to number theory
12F10 Separable extensions, Galois theory
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[1] Odoni, Mathematika 35 pp 101– (1988)
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