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Properties of iterates and composites of polynomials. (English) Zbl 0865.12003
Suppose $${\mathcal P}$$ is a property of polynomials and $$r$$ is an arbitrary natural number. This paper is concerned with the following question: does there exist a field $$K$$ and a polynomial $$f(x)\in K[ x]$$ such that the first $$r$$ iterates of $$f(x)$$ have property $${\mathcal P}$$ but the next iterate does not? (The iterates of $$f(x)$$ are defined by $$f_1 (x)= f(x)$$ and $$f_{k+1} (x)= f(f_k (x))$$ for $$k\geq 1$$.) The existence of such examples is proven for several of the most frequently considered properties of polynomials: (a) irreducibility, (b) separability, (c) splitting completely, and (d) solvability by radicals. In these examples, $$K$$ may be taken to be Hilbertian. The question of whether such examples exist over a prescribed Hilbertian field (e.g. $$\mathbb{Q}$$) is left unresolved.
Reviewer: B.Fein (Corvallis)

##### MSC:
 1.2e+06 Polynomials in general fields (irreducibility, etc.)
##### Keywords:
composition; irreducibility; polynomials; iterates; Hilbertian field
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