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