The Galois theory of iterates and composites of polynomials. (English) Zbl 0622.12011

Odoni, R. W. K.

Proc. Lond. Math. Soc., III. Ser. 51, 385-414 (1985).

Let \(F\) be a field of characteristic zero, \(f\in F[X]\) a polynomial. The iterates of \(f\) are \(f_ n(X)\), where \(f_ 1(X)=f(X)\), \(f_{n+1}(X)=f(f_ n(X))\). The generic polynomial \({\mathfrak F}\) over \(F\) of degree \(k\) is \({\mathfrak F}=X^ k+s_ 1X^{k-1}+...+s_ k\), where \(s_ 1,\dots,s_ k\) are algebraically independent over \(F(X)\). In Theorem I the author states: All \({\mathfrak F}_ n\) are irreducible over \(K=F(s_ 1,\dots,s_ k)\) and the Galois group of \({\mathfrak F}_ n\) over \(K\) is \([S_ k]^ n\), the \(n\)th wreath power of the symmetric group \(S_ k.\)
To prove this theorem it is enough to handle the case \(K={\mathbb C}(z)\); here the theory of Riemann surfaces and monodromy groups can be applied to give the result. By the same methods it is shown that the Galois group of \({\mathfrak F}({\mathfrak G}(X))\), where \({\mathfrak F}\) and \({\mathfrak G}\) are “different” generic polynomials of degrees \(k, l\), respectively, is \(S_ k[S_ l]\) (wreath product).
The results are used to study the set of primes dividing the elements of the sequence \((a_ n)\) where \(a_{n+1}=f(a_ n)\), \(a_ 0\in \mathbb Z\), and \(f\in\mathbb Z[X]\) is monic of degree \(\geq 2\) (for linear polynomials there are lots of results; cf. the paper’s bibliography): The “colloquial” version of the author in this direction is the following: “Almost all” monic polynomials \(f\in\mathbb Z[X]\) are such that the primes dividing the sequence \((a_ n)\), where \(a_{n+1}=f(a_ n)\), form a “thin set” in \(\text{spec}(\mathbb Z)\), for every choice of \(a_ 0\in\mathbb Z.\)
The paper is very carefully written and gives a lot of material partly known but here arranged in a very readable fashion.
{Reviewer’s remark: Instead of Knoblauch one should read Knobloch.}

Reviewer: K. Kiyek (MR 87c:12005)

MSC:

12F10	Separable extensions, Galois theory
12E05	Polynomials in general fields (irreducibility, etc.)
20B25	Finite automorphism groups of algebraic, geometric, or combinatorial structures

Keywords:

iterates of polynomials; composites of polynomials; field of characteristic zero; generic polynomial; Galois group; wreath power of the symmetric group; thin set