×

Galois group over \(\mathbb{Q}\) of some iterated polynomials. (English) Zbl 0758.11045

Let \(a\) be an integer such that \(-a\) is not a square in \(\mathbb{Q}\), \(f:=X^ 2+a\in\mathbb{Z}[X]\), and denote the iterates of \(f\) by \(f_ 0:=X\) and \(f_{n+1}:=f(f_ n)=f_ n^ 2+a\) for all \(n\geq 0\). Let \(c_ 1:=-a\) and \(c_{n+1}:=f(c_ n)=c_ n^ 2+a\) for \(n\geq 1\). There is an integer sequence \((b_ n)_{n\geq 1}\) with the \(b_ n\) coprime in pairs such that for all \(n\geq 1\), \(c_ n=\prod_{d\mid n} b_ d\). Let \(K_ n\) be the splitting field of \(f_ n\) over \(\mathbb{Q}\) and denote by \(\Omega_ n:=\text{Gal}(K_ n/\mathbb{Q})=\text{Gal}(f_ n/\mathbb{Q})\) its Galois group over the rational numbers. Let \([C_ 2]^ n\) denote the \(n\)-fold wreath product of the 2-element group. Then it is known that \(\Omega_ n\) always injects into \([C_ 2]^ n\). The following equivalence is shown in the paper: \(\Omega_ n\cong [C_ 2]\) if and only if \(c_ 1,c_ 2,\dots,c_ n\) are 2-independent in \(\mathbb{Q}\).
Here, “2-independent” means that no nonempty product of some of the \(c_ n\) is a square. This gives the following sufficient condition for \(\Omega_ n\cong[C_ 2]^ n\) to hold: \(| b_ m|\) is not a square for \(2\leq m\leq n\).
This condition is then verified for all \(n\) in the following cases: (\(a>0\) and \(a\equiv 1\) or \(2\bmod 4\)) or (\(a<0\) and \(a\equiv 0\bmod 4\) and \(-a\) not a square). So, for these \(a\), one always has \(\text{Gal}(f_ n/\mathbb{Q})\cong[C_ 2]^ n\) for all \(n\).
Reviewer: M.Stoll (München)

MSC:

11R32 Galois theory
11R09 Polynomials (irreducibility, etc.)
11B37 Recurrences
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. E. Cremona, On the Galois groups of the iterates ofx 2+1. Mathematika36, 259-261 (1989). · Zbl 0699.12018
[2] R. W. K. Odoni, Realising wreath products of cyclic groups as Galois groups. Mathematika35, 101-113 (1988). · Zbl 0662.12010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.