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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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##### References:
 [1] Odoni, Mathematika 35 pp 101– (1988)
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