an:04146059
Zbl 0699.12018
Cremona, J. E.
On the Galois groups of the iterates of \(x^ 2+1\)
EN
Mathematika 36, No. 2, 259-261 (1989).
00169697
1989
j
11R32 11-04 12F10
Galois groups of iterated polynomials; wreath power of finite; groups; computer program
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 \(\Omega_ n=\text{Gal}(K_ n/\mathbb Q)\). \textit{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 \(\mathbb Z/2\mathbb Z\).
T. Soundararajan (Madurai)
Zbl 0662.12010