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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)

12E05 Polynomials in general fields (irreducibility, etc.)
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